1. Don't lie. There are now unemployed looking guys at prison time for lying about VA disability to FAA.
2. Just for PITA factor, I would stay away from mental health stuff. But, take care of yourself, and refer to num 1 above. There are plenty of pilots on SSRIs, getting counseling, etc. After Germanwings there is a tad bit of hysteria about pilots lying about mental health.
3. Sleep apnea is no longer as big a deal as it once was with FAA. But you will have to travel with a CPAP, and it tells on you. If you need it for rest, by all means go for it. Just be aware of what you're signing up for.
4. If you're able to fly in AF, you shouldn't have a problem maintaining an FAA physical. Its kinda amazing what conditions that will still allow a class one.