I've rephrased the questions for mechanical clarity. I hope that's okay.
Question: If a creature has a base attack bonus of +1 and the extraordinary ability improved grab (Monster Manual 310), can the creature always take one If You're Grappling option after it starts a grapple with the improved grab ability if the creature hasn't used its base attack bonus-based attack that turn and the creature made a full attack?
Answer: Mostly yes, and this is true even if the creature's base attack bonus is +0. Further, if the creature has a base attack bonus of +6 or higher and hasn't used any of it's base attack bonus-granted attacks, it can take iterative attack-replacing If You're Grappling options. However, always is dangerous word in 3.5; better to say usually. Further, not all If You're Grappling options can be used as an attack; some are standard actions, and those can't be taken if the creature's made a full attack.
Question: If the answer to the previous question is no, then can the creature opt to suffer a −5 penalty on its natural attack and make its secondary attack first to hold onto its higher attack until after it grapples?
Answer: Luckily, it's yes, but, to be clear, a creature's natural attacks usually aren't tied to its base attack bonus at all, instead functioning independently from the creature's attacks due to its base attack bonus. The Monster Manual on natural weapons says, "The number of attacks a creature can make with its natural weapons depends on the type of the attack" (312), and not on the creature's base attack bonus. However, actually being in a grapple limits the typical creature exclusively to If You're Grappling options (and the DM's generosity), and those options are predicated on base attack bonus or the creature's ability to take standard actions. The question's difficulty lies in that it's about the transition from attacking with a natural weapon to being in a grapple.
To be clear, this is my opinion, but it's an opinion rooted in the rules. If that's enough, you needn't read the rest of this answer, but there's more detail below if you want it.
A creature that starts a grapple and has at least some its base attack bonus remaining with which to make attacks can make If You're Grappling options with those remaining attacks…
Yes, that's complicated, but, honestly, it's not my fault. What that means is that if a creature has extraordinary ability improved grab tied to a natural weapon and if that creature gets a hold using that natural weapon, if the creature has one or more attacks still available from its base attack bonus, the creature can use its remaining base attack bonus-granted attacks to take one or more If Your Grappling options that require making an attack.
I know that's not much clearer. Here's an example:
Racs, a fierce dire lion (Monster Manual 63-4), charges a gibbering mouther (126). At the charge's end, due to his extraordinary ability pounce, Racs makes a full attack. He makes one claw attack that hits and deals 4 points of damage, a second claw attack that misses, and a bite attack that hits and deals 11 points of damage. Racs extraordinary ability improved grab is tied to his bite, so he attempts to start a grapple, jumping straight to Step 3: Hold, and easily wins the opposed grapple check against the gibbering mouther.
This DM doesn't have Racs deal his unarmed strike damage because, as per the description of the improved grab ability, "A successful hold does not deal any extra damage unless the creature also has the constrict special attack" (310), and this DM considers unarmed strike damage extra damage when establishing the hold with improved grab; another DM's opinion may vary. To continue the example:
So far this turn Racs hasn't used his base attack bonus at all. That means he can, because of his base attack bonus +8, pick two If Your Grappling options that count as attacks. For his first attack at his full base attack bonus, Racs picks Damage Your Opponent, makes an opposed grapple check, wins against the mouther, and deals the mouther 8 points of nonlethal damage. For his second attack at his base attack bonus −5, Racs picks Damage Your Opponent, makes an opposed grapple check with his check suffering a −5 penalty, still wins against the mouther, and deals the mouther 9 points of nonlethal damage.
This DM wouldn't have Racs use the grapple option Attack Your Opponent with his natural weapons this turn: Racs has used his natural attacks already, and letting Racs make attacks with them twice seems, to this DM, unfair. (Racs may disagree.) In fact, even next turn, this DM probably wouldn't have Racs take the Attack Your Opponent option and, for instance, have Racs both times use his bite attack. The rules may permit it, but it still makes this DM uneasy. Were the creature to have a lone natural weapon, this DM would consider that, but even then he'd probably have the creature take the Damage Your Opponent option for its second and later iterative grapple options.
Nonetheless, a literal reading could see Racs's full attack be a claw attack, a claw attack, a bite attack that via the improved grab ability starts a grapple, then, if the grapple's successful, the If You're Grappling option Attack Your Opponent at Racs's full base attack bonus with a claw at a −4 penalty due to the penalty applied by that option, and finally the If You're Grappling option Attack Your Opponent again at Racs's base attack bonus −5 (his iterative attack) still at a −4 penalty. To this DM, that seems not just wrong but excessive, but if trying to maximize a creatures' damage output, this may be what a dire lions does.
…Even if the creature's already made attacks with natural weapons
To be extra clear, as Racs the dire lion demonstrates, this DM doesn't have a creature suffer a penalty on its grapple checks (or its attack rolls made while in a grapple) for a creature having already made this turn attacks with its natural weapons. While the Player's Handbook on Full Attack says, "If you get multiple attacks because your base attack bonus is high enough, you must make the attacks in order from highest bonus to lowest" (143), natural attacks are usually all made at a creature's highest base attack bonus—even a creature's attacks with its secondary natural weapons are still made at the creature's highest base attack bonus, albeit typically with −5 penalty. This means a creature can employ its natural attacks in any combination and even, if capable, make its highest base attack bonus-derived attack right alongside its natural attacks. (Contrast this with iterative attacks derived from a creature's base attack bonus: those iterative attack aren't suffering a penalty when they're made but instead are made at an actual lower base attack bonus: "Numbers after a slash [in base attack bonus] indicate additional attacks at reduced bonuses" (PH 22) rather than, for example, at penalties.)
Also, it seems to this reader that this penalty for making secondary natural attacks doesn't also apply to attacks derived from the creature's base attack bonus. (The Monster Manual on Natural Attacks, in part, says, "An attack with a primary natural weapon uses the creature’s full attack bonus. Attacks with secondary natural weapons… are made with a −5 penalty on the attack roll…" (312), but this is largely in the context of the creature's typical full attack. By ruling that this general case overrides the specifics of If Your Grappling, Racs, for instance, could make his two base attack bonus-derived claw attacks using his full base attack bonus, an odd and—so far as I'm aware—uniquely powerful situation.)
Further, If You're Grappling says
When you are grappling (regardless of who started the grapple), you can perform any of the following actions. Some of these actions take the place of an attack (rather than being a standard action or a move action). If your base attack bonus allows you multiple attacks, you can attempt one of these actions in place of each of your attacks, but at successively lower base attack bonuses. (156)
…And this DM reads this as a creature's previous use of its natural weapons—primary or secondary—in the same turn as not affecting the creature's base attack bonus. A creature like Racs—that typically doesn't employ its base attack bonus during its turn at all—has its full base attack bonus remaining when it enters a grapple, so it "can perform any of the following [If You're Grappling] actions [if it has sufficient actions remaining, and s]ome of these actions take the place of an attack" that's derived from the creature's base attack bonus, all of which Racs typically has. (N.b. A creature like Racs can take the full attack action and use its base attack bonus to make unimproved unarmed strikes—each strike provoking an attack of opportunity from its foes—, either before, during, or after its natural attack routine. However, this DM only has truly desperate creatures engage in such tactics. So, for example, while it's rare, in this DM's campaigns sometimes a PC gets headbutted repeatedly by a bear.)
If a creature's already made an attack with its primary natural weapon this turn, a DM may nonetheless assess during a grapple a −5 penalty on all the creature's natural weapon attacks or grapple checks with the claim that the creature, already having attacked with its primary weapon, has made all of its ensuing attacks—whether base attack bonus-derived or not, whether primary or secondary—essentially secondary. This reader can't stop that DM, but to this reader that seems overcomplicated and also seems to overapply the general case of the creature's normal full attack routine with its natural weapons to the specific case of the If Your Grappling rules, and those rules don't mention doing this. Still, this player likely wouldn't leave a DM's typical campaign over this ruling, but in an E6 campaign, for instance, where my PC could specialize in natural weapons and grappling and where my PC didn't have to worry about late-game monsters routinely obviating the latter, I'd definitely be annoyed by it.
By now it should be obvious that the process the question asks about is never actually described in the rules. This reader and DM has extrapolated this chain of events from the existing rules as best he could, often as outgrowths of conversations with players whose PCs were grappled or whose minion-monsters were grappling. Further, Dungeons & Dragons 3.5 has remarkably few step-by-step examples of combat that can be marshaled to determine how this is supposed to work. Heck, even one of the game's original designers had one of his online columns on this topic—in a rare move—subjected to errata (cf. Skip William's Rules of the Game Web column "All about Grappling (Part Four)").
In other words, if the process described above doesn't work for your group, official word is unlikely forthcoming over a decade after the game's ceased publication. Talk to your group, agree on how it does work, and run with it.