If I get a care package, which contains a spy plane, does using that spy plane count towards my spy plane killstreak challenges?
Yes. It does, all killstreak rewards that came from the care packages count to your killstreak challenges. In fact, I had a Chopper Gunner from a care package which was added to my Chopper Gunner killstreak count. Which proves killstreak rewards earned from care packages definitely count towards to the killstreak challenges.