The only mouse controls for airplanes are for free look. The flight controls are strictly keys.
The best thing I have found is to cut the throttle (hold C) until it is so low you begin to loose altitude in level flight. Then I hit up throttle (tap Shift) once. It is just enough to keep you from losing altitude and when you go nose up the throttle auto manages itself around the throttle setting you have. This allows you fly very slowly, like STOL plane, yet quickly nose up to climb out of trouble without changing settings. Gives you much more time for strafing and lining up bomb runs. For dog fights a similar approach works except at about 50% throttle. At full throttle you will fly past the opponent and put yourself into his gun sights. At half you will be in the same speed ballpark as the enemy. The plane is easier to fly at reduced speeds, though I say that loosely.
If it will help I could do a quick 'how-to' video for keybrd and mouse flight in airplanes. I don't claim to be any great shakes though I think I have a handle on it.
The big problem is (I think) we are losing dev's with broad keyboard and mouse experience as time goes on. I keep seeing the same mistakes that were made early on in PC gaming, being repeated again today. The dev's really need to go back and look at preconsole K&M games to see more detail on 'standard' K&M control setups. Or ask old gray beards like me or others in the community to help with conceptual design on the K&M setups maybe. If they had just given use the Classic setup, it would work fine, but they cheaped out on that aspect of the PC port.
Took me quite a while to beat Spray and Pray, but it was also because I made the mistake of playing this Clutch Nixon mission before any other, and in the semi dark hours of early evening.
What finally helped me get through it was, 1. inverting plane controls so W is up and S is down, 2. using max FOV, 3. setting look sensitivity to 50%. I also tend to slowly tap W as I turn, as planes tend to drop altitude when you turn, which is realistic as I've actually flown a small sea plane before. It was part owned with 5 other guys by a neighbor whom I helped wash n wax it once. In return he took me on a short flight and taught me some things and let me fly it a bit. Also, make sure you line up well in advance, like sweeping wide first before making a long arcing turn.
The most difficult part of this course for me was definitely the tight right/left S turn about 1/3 way through. Everything past that has pretty good line of sight, much less aggressive turns, and rings you can see well in advance. Another tip is if you can afford it, skip a hard ring and compensate by picking up speed in the sections where the rings are aligned in more straight fashion. You don't have to hit all the rings, they just add more time.
Anymore I can pretty much nail the flight stunt courses in a few tries, save for this one which would probably take at least several. As long as I try them at best light and can see well, the control part of it feels more intuitive now, even without rebinding any keys. Come to think of it there was one other one in Henbane region that had S turns through a tight canyon that would probably also take several tries.
What was actually more frustrating than the plane courses or controls was ShadowPlay's Instant Replay having a bug where the IR clips freeze and pixelate just over halfway through the playback, so I never got a playable capture file of any of the stunts. Hopefully now that Nvidia has finally acknowledged a problem with ShadowPlay in their latest driver release notes, it will be fixed by the time I do my Hard play through. These are not the type of missions you want to record with manual capture, as they can take many tries.