Is the Flying-type essential to CAP37's success? What angles are there for a non-Flying-type that regularly makes use of Beak Blast?
It's... complicated. The Flying type famously lacks reliable high-power moves, to the point that defensive Pokémon like Corviknight run Brave Bird. So if a reliable Physical Flying-type STAB is provided, I think the reliable power would be a stronger reason to use the move than its conditional effect, unless the drawbacks of getting hit/negative priority contrast with the Pokémon's role. But Flying also invites Knock Off, which could be a great way to play into the conditional aspect (but probably only once, and assuming the entire typing is still Rocks-weak).
If we make it non-STAB, we do not run the risk of Beak Blast only being used for its power, and have some more space to explore the conditional effect. Another "advantage" of non-STAB is that Flying already resists three typically Physical types, and is not in need of too much helping, but Beak Blast could output more value (with the Burn) on something typing-wise more physically vulnerable. Ting-Lu was mentioned by spammernoob on Discord as a Pokémon that might enjoy Beak Blast, given its monstrous Special bulk, weaknesses to Water, Grass, Fighting, and Bug, which are common physical attacking types. Or a Psychic type that doesn't have to fear Knock Off or U-turn as much. Not being limited to Flying thus gives us more flexibility to tailor our dynamics with contact moves, whereas Flying puts an inherent limiter by being p. good against three already.
Also: is Flying really a bad coverage type, or is it that Flying options are generally terrible? This could be an interesting aspect to explore with Beak Blast. This also allows us to diversify our role from the proven bulky Flying angle in STABmons, where we could instead, say, focus on a bulky wallbreaker who doesn't mind going last with huge offenses and for whom Flying coverage is good. Scizor runs Dual Wingbeat sometimes, Pajantom Brave Bird, but not many examples. Acrobatics saw use on Roaring Moon, but that was for the synergy with Protosynthesis and the immediate power to make use of a +1 Atk or Speed. Apples and oranges to the slower Beak Blast, so this really is somewhat unexplored space.
What examples exist in other metagames or generations that we can learn from what makes offensive Pokemon succesful while using Flying-type moves? Flying has a neutrally-inclined presense on the type chart, resisted by three and super effective on three, what 'mons successfully use Flying-type moves offensively and why?
Strong neutral coverage, no immunities, not for secondary effects. Normally, the examples I've seen are just big, overwhelming damage. Both Venomicons, Cawmodore, Tornadus-therian, and the Kanto birds. But as some others have mentioned, the lack of a secondary effect in most cases also makes defensive Pokémon prefer a move with an additional effect. Beak Blast gives us that, so it's a more interesting space to explore this question.
What is the significance of the burn effect part of Beak Blast? Are there types for which this effect comes into play more than often? Is there something to consider in Fire-types being unable to get burned that plays into how CAP37's typing impacts its play?
Flying, as mentioned already, has great reinforcing synergy due to the Knock/Heavy-Duty-Boots dynamic and resisting types that are more commonly Physical. However, it also comes with an interesting mechanical wrinkle. Assuming we don't patch up the SR weakness, we can't easily switch into Knockers. Unlike contact-abilities, the use of Beak Blast is heavily telegraphed, so we cannot threaten catching moves on a switch. If we somehow enter the battle into a Knocker, we have to consider the very real possibility of them switching out on our Beak Blast into a better matchup. This idea of a "free" turn, however, could also be space for us to do other things, or create lose-lose situations for opponents.
I'm not sure if, say, resisting Knock, U-turn, or Flip Turn is better for this counterplay dynamic (as why would our opponent not simply switch out here?), or if Beak Blast can function as a deterrent to these moves on a Pokémon typically weak to them and thus patch up some of its weaknesses.
Beak Blast only deals damage at the (usual) end of the turn, with a -3 Priority to land assuming CAP37 survives the turn. Does this impact the types chosen at all, for example are typings with useful resistances more preferential to increase likelihood of survival?
Typing could definitely help with the survival aspect, but the opposite could be true too: a weaker typing to Physical moves could be improved by the mere presence of Beak Blast. Depends on whether we want to double down (which also encourages switching out by opponents), or patch up.