Confirm the DM's rulings
To do this in the first place requires that the DM rule that the variant paladin of tyranny (Unearthed Arcana 53–5) can have the ashworm that's gained from the prestige class ashworm dragoon (Sandstorm 66–70) become the paladin of tyranny's mount by the paladin of tyranny performing a ritual like the Consecration of the Sifting Sand (Sa 67), just like a typical Player's Handbook paladin could. This wouldn't normally be that big of a deal, but the typical paladin's Consecration ritual makes that typical paladin's ashworm nonvenomous—a blow that weakens some traditional ashworm dragoon class features. I recommend the player and DM reach some compromise that allows the Kwisatz Haderach paladin of tyranny/ashworm dragoon to keep his sandashworm well and thoroughly envenomed.
Answers
The question has five distinct parts. If interested only in answers to those, read on.
Yes. A paladin of tyranny 10/ashworm dragoon 10 that applied the template warbeast to her ashworm mount would have a special mount ashworm with, effectively, 18 Hit Dice:
- 3 base Hit Dice via the typical ashworm (Sa 140)
- +1 Hit Die via the template warbeast (Monster Manual II 219)
- +4 bonus Hit Dice via the paladin of tyranny 10 spell-like ability special mount
- +10 bonus Hit Dice via the prestige class ashworm dragoon extraordinary ability ashworm mount
While the paladin of tyranny/ashworm dragoon can just write into her background that her ashworm received warbeast rearing and training when it was just a larva, if she subsequently loses that ashworm, acquiring another warbeast-reared-and-trained one will be ardous. To have an ashworm acquire the template warbeast the ashworm "must be reared for one year…, then trained for 2 months…" (MM2 219).
Thus this player suggest also including in the paladin/dragoon's background at least partial ownership (perhaps as a reward for her evil deeds) of an ashworm farm where ashworms are, in the background, constantly being reared and trained as warbeast ashworms… just in case. (This is a level 20 PC we're talking about, after all!)1
Usually no. As bonus Hit Dice—like those gained by an animal companion or special mount, where the general rules for bonus Hit Dice are described—only do what they say they do, bonus Hit Dice see creatures gain extra hp, "improve [their] base attack and base save bonuses[,] and gain… additional skill points [and] feats" (PH 36, 45), so bonus Hit Dice typically don't increase creatures' sizes according to their Advancement entry in their stat block nor do bonus Hit Dice, for example, in and of themselves, improve creatures ability scores. (This is confirmed for druids and their animal companions' bonus Hit Dice specifically by the FAQ (14)… for whatever that's worth.) (Also see this question.)
An individual DM may, of course, rule otherwise, allowing the ashworm mount to see its size increase due to its bonus Hit Dice as per its Advancement entry.2 A player may find his case strengthened by pointing out that, while the ashworm dragoon grants her ashworm mount bonus Hit Dice, unlike the druid's animal companion and the paladin's special mount, the terse description of the extraordinary ability ashworm mount omits any description of what those bonus Hit Dice actually grant! Sincerely, this fellow player wishes you luck.
Finally, no matter the DM's ruling, the 1 Hit Die gained as a result of the template warbeast is a for-reals Hit Die, doing everything a Hit Die should… including, for example, when an ashworm sees its actual Hit Dice go from 3 to 4, granting it an ability score increase.
The Monster Manual Table 4–1: Changes to Statistics by Size (291) (also the SRD) details the adjustments the DM makes when a creature increases in size due to advancing Hit Dice. If the DM makes a house rule that says Bonus Hit Dice increase a creature's size as per its stat block, that DM should use these modifiers.
The ashworm's saving throws are based on its Hit Dice (including bonus Hit Dice as in the first answer, above). Because the ashworm possesses the type magical beast, its good saving throws are Fort and Ref with Will being poor. This is the same saving throw advancement as the ranger, so to determine the ashworm's base saving throw bonuses only, treat the ashworm as a ranger with levels equal to its Hit Dice.
However, a paladin's special mount ability share saving throws (PH 45) allows the paladin's mount, if the paladin's base saving throw is better than the mount's base saving throws, to substitute her base saving throw for the mount's. (In short, the game doesn't want the paladin's mount, for example, dying instantly to stuff the paladin herself would survive!)
Yes. Bonus Hit Dice explicitly count for determining skill points and feats. Note, however, that the DM creates, controls, and determines the results of advancement of NPCs (Dungeon Master's Guide 4), and that includes the ashworm dragoon's ashworm.
That said, here's the most restrictive paladin or tyranny 10/ashworm dragoon 10's ashworm special mount stat block:
Poly the ashworm
Poly Chaete, ashworm mount special mount warbeast ashworm: CR —; Large magical beast [10 ft./5 ft.]; HD 18d10+90; hp 189; Init +0 (+0 Dex); Spd 50 ft., burrow 30 ft., climb 5 ft.; AC 32 (−1 penalty via size Large +0 Dex +23 natural armor bonus), flat-footed 32, touch 9; base atk bonus +18, grp +31; atk or full atk +18 base +9 Str +1 bonus via the feat Weapon Focus sting) −1 penalty via size Large equals +27 melee (1d8+13, sting); SA poison; SQ combative mount, darkvision 60 ft., empathic link, evasive diver (largely obviated by improved evasion), improved evasion, share saving throws, share spells, tremorsense 60 ft.; SV normally Fort +16 (+11 base +5 Con), Ref +13 (+11 base +0 Dex +2 bonus via the feat Lightning Reflexes), Will +11 (+6 base +3 Wis +2 bonus via the feat Iron Will) but, if better, use the paladin's base saving throw and the ashworm's ability modifier via the special ability share saving throws; AL N; Str 28, Dex 10, Con 20, Int 7, Wis 16, Cha 6.
Skills and Feats: Appraise (Int) −2 [0 ranks −2 Int], Autohypnosis (XPH 36) (Wis) — [0 ranks +3 Wis], Balance (Dex) +0 [0 ranks +0 Dex], Bluff (Cha) −2 [0 ranks −2 Cha], Climb (Str) +17 [0 ranks +9 Str +8 racial bonus], Concentration (Con) +5 [0 ranks +5 Con], Control Shape (MM 303) (Wis) — [0 ranks +3 Wis], Craft (any) (Int) −2 [0 ranks −2 Int], Decipher Script (Int) — [0 ranks −2 Int], Diplomacy (Cha) −2 [0 ranks −2 Cha], Disable Device (Int) — [0 ranks −2 Int], Disguise (Cha) −2 [0 ranks −2 Cha], Escape Artist (Dex) +0 [0 ranks +0 Dex], Forgery (Int) −2 [0 ranks −2 Int], Gather Information (Cha) −2 [0 ranks −2 Cha], Handle Animal (Cha) — [0 ranks −2 Cha], Handle Humanoid (April Fools column “Fabulous Cats!”) (Cha) — [0 ranks −2 Cha], Heal (Wis) +3 [0 ranks +3 Wis… and, no, I don't know either], Hide (Dex) −4 [0 ranks +0 Dex −4 penalty via size Large], Iaijutsu Focus (OA 58-9) (Cha) −2 [0 ranks −2 Cha], Intimidate (Cha) −2 [0 ranks −2 Cha], Jump (Str) +17 [0 ranks +9 Str +8 bonus via speed 50 ft.], Knowledge (all) (Int) −2 but maximum result 10 [0 ranks −2 Int], Listen (Wis) +27 [21 ranks +3 Wis +2 bonus via the feat Alertness +1 racial bonus], Lucid Dreaming (MP 203) (Wis) — [0 ranks +3 Wis], Martial Lore (ToB 28) (Int) — [0 ranks −2 Int], Move Silently (Dex) +0 [0 ranks +0 Dex], Open Lock (Dex) — [0 ranks +0 Dex], Perform (any) (Cha) −2 [0 ranks −2 Cha], Profession (all) (Wis) — [0 ranks +3 Wis], Psicraft (XPH 38) (Int) — [0 ranks −2 Int], Ride (Dex) +0 [0 ranks +0 Dex], Search (Int) −2 [0 ranks −2 Int], Sense Motive (Wis) +3 [0 ranks +3 Wis], Sleight of Hand (Dex) — [0 ranks +0 Dex], Speak Language (Common but can’t speak) [0 ranks], Spellcraft (Int) — [0 ranks −2 Int], Spot (Wis) +6 [0 ranks +3 Wis +2 bonus via the feat Alertness +1 racial bonus], Survival (Wis) +3 [0 ranks +3 Wis], Swim (Str) +9 [0 ranks +9 Str], Truespeak (TM 195-6) (Int) — [0 ranks −2 Int], Tumble (Dex) — [0 ranks +0 Dex], Use Magic Device (Cha) — [0 ranks −2 Cha], Use Psionic Device (XPH 38–9) (Cha) — [0 ranks −2 Cha], Use Rope (Dex) +0 [0 ranks +0 Dex]; Alertness (PH 89), Improved Natural Attack (sting) (MM 304), Iron Will (PH 97), Lightning Reflexes (PH 97), Power Attack (PH 98), Run (PH 99), Weapon Focus (sting) (PH 102).
Combative Mount (Ex): A rider astride you gains a +2 bonus on Ride skill checks made regarding you. You're proficient with light, medium, and heavy armor.
The ability score increase from the Hit Die granted by the template warbeast has been applied to Poly's Wisdom.
Here's the least restrictive:
Pria the ashworm
Pria Pulid, ashworm mount special mount warbeast ashworm: CR —; Huge magical beast [15 ft./10 ft.]; HD 18d10+144; hp 243; Init −1 (−1 Dex); Spd 50 ft., burrow 30 ft., climb 5 ft.; AC 33 (−2 penalty via size Large −1 Dex +26 natural armor bonus), flat-footed 32, touch 7; base atk bonus +18, grp +40; atk or full atk +18 base +14 Str −2 penalty via size Huge equals +30 melee (1d6+21 plus poison, sting); SA poison (Fort DC 27; initial and secondary damage: 2d6 Str); SQ combative mount, darkvision 60 ft., empathic link, evasive diver (largely obviated by improved evasion), improved evasion, share saving throws, share spells, tremorsense 60 ft.; SV normally Fort +19 (+11 base +8 Con), Ref +12 (+11 base −1 Dex +2 bonus via the feat Lightning Reflexes), Will +9 (+6 base +3 Wis) but, if better, use the paladin's base saving throw and the ashworm's ability modifier via the special ability share saving throws; AL N; Str 38, Dex 8, Con 26, Int 7, Wis 15, Cha 6.
Skills and Feats: Appraise (Int) −2 [0 ranks −2 Int], Autohypnosis (XPH 36) (Wis) — [0 ranks +2 Wis], Balance (Dex) −1 [0 ranks −1 Dex], Bluff (Cha) −2 [0 ranks −2 Cha], Climb (Str) +22 [0 ranks +14 Str +8 racial bonus], Concentration (Con) +8 [0 ranks +8 Con], Control Shape (MM 303) (Wis) — [0 ranks +2 Wis], Craft (any) (Int) −2 [0 ranks −2 Int], Decipher Script (Int) — [0 ranks −2 Int], Diplomacy (Cha) +19 [21 ranks −2 Cha… because it can and it's hilarious], Disable Device (Int) — [0 ranks −2 Int], Disguise (Cha) −2 [0 ranks −2 Cha], Escape Artist (Dex) −1 [0 ranks −1 Dex], Forgery (Int) −2 [0 ranks −2 Int], Gather Information (Cha) −2 [0 ranks −2 Cha], Handle Animal (Cha) — [0 ranks −2 Cha], Handle Humanoid (April Fools column “Fabulous Cats!”) (Cha) — [0 ranks −2 Cha], Heal (Wis) +2 [0 ranks +2 Wis… and, no, I still don't know], Hide (Dex) −9 [0 ranks −1 Dex −8 penalty via size Huge], Iaijutsu Focus (OA 58-9) (Cha) −2 [0 ranks −2 Cha], Intimidate (Cha) −2 [0 ranks −2 Cha], Jump (Str) +22 [0 ranks +14 Str +8 bonus via speed 50 ft.], Knowledge (all) (Int) −2 but maximum result 10 [0 ranks −2 Int], Listen (Wis) +3 [0 ranks +2 Wis +1 racial bonus], Lucid Dreaming (MP 203) (Wis) — [0 ranks +2 Wis], Martial Lore (ToB 28) (Int) — [0 ranks −2 Int], Move Silently (Dex) −1 [0 ranks −1 Dex], Open Lock (Dex) — [0 ranks −1 Dex], Perform (any) (Cha) −2 [0 ranks −2 Cha], Profession (all) (Wis) — [0 ranks +2 Wis], Psicraft (XPH 38) (Int) — [0 ranks −2 Int], Ride (Dex) −1 [0 ranks −1 Dex], Search (Int) −2 [0 ranks −2 Int], Sense Motive (Wis) +2 [0 ranks +2 Wis], Sleight of Hand (Dex) — [0 ranks −1 Dex], Speak Language (Common but can’t speak) [0 ranks], Spellcraft (Int) — [0 ranks −2 Int], Spot (Wis) +3 [0 ranks +2 Wis +1 racial bonus], Survival (Wis) +2 [0 ranks +2 Wis], Swim (Str) +14 [0 ranks +14 Str], Truespeak (TM 195-6) (Int) — [0 ranks −2 Int], Tumble (Dex) — [0 ranks −1 Dex], Use Magic Device (Cha) — [0 ranks −2 Cha], Use Psionic Device (XPH 38–9) (Cha) — [0 ranks −2 Cha], Use Rope (Dex) −1 [0 ranks −1 Dex]; Improved Bull Rush (PH 95), Martial Study (the 1st-level White Raven martial maneuver douse the flames [strike] (ToB 91)) (ToB 31–2), Martial Study (the 3rd-level White Raven martial maneuver white raven tactics [special] (ToB 94)) (ToB 31–2), Power Attack (PH 98), Shock Trooper (CW 112), and either Blowhard (SS 31) or even Magic Devotion (CC 61) if a ranged effect is desired or Improved Unarmed Strike (PH 96–7) if melee improvement is desired.3
Combative Mount (Ex): A rider astride you gains a +2 bonus on Ride skill checks made regarding you. You're proficient with light, medium, and heavy armor.
The ability score increases from changing Pria's size from Large to Huge have already been applied. Because this DM ruled that bonus Hit Dice are treated as normal Hit Dice, in addition to the ability score increase gained from the 1 Hit Die added by the template warbeast, Pria also saw ability score increases when it reached 8, 12, and 16 Hit Dice. Ability score increases were split evenly between Str and Con so that its Str modifier remained an even number.
1 A typical ashworm costs 200 gp (Sa 103) but a typical warbeast ashworm costs only 150 gp. As this is nonsense—and the warbeast template as written creates a pretty crazy world anyway—, I suspect a DM to rule buying warbeast ashworm a far more difficult task than just heading to the local village and handing 150 gp to the local ashworm handler.
2 The ashworm's Advancement entry is 4–6 HD (Large); 6–9 HD (Huge). Thus, under such house rules, the DM must determine, when an ashworm is advanced to 6 Hit Dice, whether it stays Large or becomes Huge.
3 Even with the feat Improved Unarmed Strike the creature still suffers a −4 penalty on its attack rolls due to nonproficiency with its unarmed strike—the feat, oddly, not actually granting proficiency with the simple weapon that is an unarmed strike—, but the creature won't provoke attacks of opportunity with its unarmed strike and can opt to deal lethal or nonlethal damage. More importantly, the feat permits the creature to use its base attack bonus for iterative attacks (hence its sting as a secondary natural attack) and apply to the unarmed strike the benefit of the feat Power Attack, unlike, for example, the light weapon gained via a feat like Martial Weapon Proficiency (spiked armor) (PH 97). Pria taking the feat Improved Unarmed Strike also encourages her rider to, if allowed, share with Pria the spell the rider casts from a wand of lion's charge [trans] (SpC 133) (2nd-level spell at caster level 4) (120 gp/charge), likely, however, requiring the rider to invest ranks in the skill Use Magic Device.