When activating the ability on Panic Spellbomb (tap, Sacrifice Panic Spellbomb: Target creature can't block this turn.), can the opponent then use an Instant such as Banishing Stroke to put Panic Spellbomb on the bottom of it's owner's library? Both are happening at instant speed, so do they stack like typical instants?
Your opponent can't do that. Sacrificing the Spellbomb is part of the cost, and you pay costs as part of activating the ability and putting it on the stack, and only once you're done with all that does anyone get priority. So by the time your opponent gets the chance to do anything, it's already in your graveyard. (This is just like how your opponent couldn't do something in response to you paying mana for a spell or ability.)
If the Spellbomb weren't being sacrificed, your opponent could cast Banishing Stroke on it, but it wouldn't really accomplish anything. Once an ability is activated, it exists on the stack independently of its source. (Yes, activated abilities use the stack.) You'd activate the ability, and your opponent could cast Banishing Stroke in response. The Banishing Stroke would then be on top of the stack, and resolve first. It'd put the Spellbomb on the bottom of your library, then the Spellbomb's ability would resolve as usual.
Side note: all spells and activated abilities use the stack, regardless of whether they're instants (or at "instant speed"). The important thing about instants and activated abilities is that they can be cast/activated any time you have priority, e.g. while something's already on the stack. That is, the bottom thing on the stack definitely doesn't have to be at "instant speed", just the things added on top of it (usually) - and the stack resolves top-down the same way no matter what's on it.
All spells, activated abilities, and triggered abilities use the stack. Panic Spellbomb's ability is no exception. Although it does happen at instant speed, Panic Spellbomb's ability is not actually an instant spell, it is an activated ability. From the comprehensive MTG rules:
602.1. Activated abilities have a cost and an effect. They are written as "[Cost]: [Effect.] [Activation instructions (if any).]"
Panic Spellbomb's ability reads: "Tap, Sacrifice Panic Spellbomb: Target creature can't block this turn." Sacrificing Panic Spellbomb is part of the cost of activating it's ability because it comes before the colon.
602.2. To activate an ability is to put it onto the stack and pay its costs, so that it will eventually resolve and have its effect.
This means that in order to put Panic Spellbomb's ability on the stack, you must pay its costs (tap it and sacrifice it) first, meaning that it would be in your graveyard before it even went on the stack in the first place. Panic Spellbomb would then be an illegal target for Banishing Stroke as Panic Spellbomb would be in the graveyard.