Many, if not most, property runes don't work
If you take on a battle form with a polymorph spell, the special statistics can be adjusted only by circumstance bonuses, status bonuses, and penalties. Unless otherwise noted, the battle form prevents you from casting spells, speaking, and using most manipulate actions that require hands. (If there's doubt about whether you can use an action, the GM decides.) Your gear is absorbed into you; the constant abilities of your gear still function, but you can't activate any items.
Runes that function through activations like earthbinding and extending don't work, because you can't activate them.
For runes that add extra damage to attacks we can get a solid clue elsewhere. Namely, dragon transformation, which states:
You transform into a ferocious Large dragon, gaining the effects of 6th-level dragon form except that you use your own AC and attack modifier, you apply your extra damage from Rage, and the Breath Weapon uses your class DC.
This feat basically confirms that extra damage is an adjustment of the form's statistics, or else you'd be able to add your rage damage to the dragon form even without it. Property runes like shock or disrupting which increase your damage likewise do so via extra damage, not via circumstance or status bonuses, so their damage does not work.
And as you've already pointed out, lines like "plus 1d6 fire" that are found in some battle forms also support this interpretation.
The fate of the other property runes is more ambigous. The notion that "the special statistics can be adjusted only by circumstance bonuses, status bonuses, and penalties" could be reasonably be refering to numerical statistics only, because all the adjustments that do work are numerical too. In this interpretation the rest of the property runes by and large work with the battle forms.
However, a GM could reasonably rule that adjustments to any kind of statistic must be circumstance or status bonuses. The statistics of battle forms are clearly similar in format to creature statistics, especially the attacks, so it wouldn't be a stretch to read the form's attacks through the lens of Reading Creature Statistics:
Melee [A] (traits; some weapon traits, such as deadly, include their calculations for convenience) The name of the weapon or unarmed attack the creature uses for a melee Strike, followed by the attack modifier and traits in parentheses. If a creature has any abilities or gear that would affect its attack modifier, such as a weapon with a +1 weapon potency rune, those calculations are already included, Damage amount and damage type, plus any additional effects (this entry is Effect if the Strike doesn’t deal damage).
Ranged [A] As Melee, but also lists range or range increment with traits, Damage as Melee.
Here we can see that everything from the traits to the additional effects are part of a creature's attack statistic, and this is probably true for battle form statistics too. Just like with extra damage, we can find circumstancial evidence for this in some battle forms. For example, some of the aberrant form's attacks say "and you can spend an action after a hit to Grab the target", one of the choral angel form's attack says "deafened for 1 round on a critical hit", the moon cosmic form says "on a critical hit with either Strike, the target is stupefied 2 for 1 round", etc.
So if the GM rules that non-numerical statistics must also be adjusted by circumstance or status bonuses, then property runes which change weapon traits like merciful and hooked, and property runes which grant additional effects to attacks such as ghost touch and (the on crit effects of) distrupting do not work for battle forms.
In short, activations and extra damage don't work. Trait changes and additional effects are up to GM interpretation. I believe those 4 categoriers catch almost every property rune, with these few exceptions.
- Authorized. Does anyone even care about this rune?
- Ancestral echoing. The extra lore just works. Every form I've checked is trained in their attacks, and this rune could make you expert if the GM allows non-numerical adjustments, but you can't actually increase the form's attack modifier because that would definitely be an adjustment that doesn't use circumstance or status bonuses.
- Speed. The quickened condition just works because it has nothing to do with the battle form's statistics.
- Impossibile. The protection against dispell magic and similar effects works because it protects the handwraps and has nothing to do with the battle form's statistics, though I'm not sure the handwraps could be targetted by such effects while they are absorbed into you. The adjustment to range follows the same logic described for traits.