The rules as written angle have been handled, but the idea got me googling. Since you didn't actually specify rules as written vs lore or whatever, here's some info and thoughts on real world birds to give you somewhere to start. You can use this to justify or handwave your character, or for roleplaying ideas.
Real world bird respiration, it turns out, is pretty weird. They have relatively small lungs, then extra air sacs in various places. The lungs actually empty when inhaling, and fill when exhaling. I have no idea what effect this would have on woodwind instruments, and of course there's no indication Aarakocra work the same way.
In any case, from what I can google, birds can in fact breath through their noses. A bird-person could potentially play a human-style recorded by wrapping the mouthpiece in a strip of cloth to act like a gasket, cram the thing in one nostril, and plug the other.
Transverse flutes--the kind you hold to the side--are likely a different story. They're played by shaping the lips to blow at a particular angle over an opening. While the corners of birds' mouths can stretch, as evidenced by the various comical gull-eating-a-starfish photos strewn across the internet, I can't find any indication of whether birds have muscles that can shape these, and even if they can it's hard to imagine the rest of the beak seals well enough for it to matter. It's possible they could hold the flute at a funny angle and get the airflow right, but it'd likely be very difficult to play. Again, real world anatomy may not apply, but it could make for some interesting roleplay.
As for your last question, not all birds can produce musical vocalizations. You'd be hard-pressed to caw an aria or quack a ballad. This can be used to justify needing or not needing an instrument at your discretion. Even if they can sing musically, various implements could be used to alter or amplify the sound, like how the musician provides the buzz for a trumpet.