Flute questions (NOT RECORDER)?
I have been a professional flutist since 1973 (after first 2 degrees - a third followed.) That means I was a STUDENT taking lessons for many years before that. I qualified, degreed, experienced TEACHER can fix you in a couple of lessons - and then you will be eager to see what ELSE they can help you with. The embouchure required for flute in critical - and unlike any other woodwind. Unless you are taught that by a flutist (not a clarinet or sax player!!!) then you are doomed. As far as recording - the placement and quality of the mic is important. I have tried quite a few over the decades - and frankly, to go old-school, the one that f gave me the LEAST wind noise was the Barcus Berry that replaced the cork assembly in my headjoint. Yes, it was wired and had small floor box that ran to the mixed - but it worked great. Some players use Countryman mikes - some use head mikes, the same as singers. Air noise is inherent to flute playing - as string slide noise is to acoustic guitarists. But since you say that you are having f difficulty with notes ( that should actually be fairly easy to get clearly, especially the F and ag - the E can be a stinker) then this goes back to my first suggestion - find a really good teacher. They will also check over your flute, and see if there are issues there as well. You sound serious about all this, so I hope that you follow our advice here - if you could do it ALONE, then there would be no need for teachers - and there are many thousand of us out there who are booked solid, so obviously we are a big help to folks who need it. Good luck - start looking for one of my colleagues!
The playing problems you describe would be best answered by a qualified teacher who can actually see and hear you play. If you don't have one then I guess that's the reason for your problems. So far as being able to hear your breathing when you record yourself, that's a problem with your recording technique. Everyone breathes when playing and breathing makes a sound. The trick is to avoid picking up that soubd on the microphone. Again, without knowing more about what you are doing it's impossible to help. Obviously don't have the microphone too near to or pointed directly at your mouth.