OSA is a separate issue, there are threads on that here that you can look at.
Anxiety/depression could take a while. Have you ever been diagnosed or talked about it with any health care provider? Sounds like you probably did, so you'll have to report those visits to the FAA on the medical form. That includes non-MD counselors (exception for marriage/family counseling IIRC).
With depression/anxiety, any suspicion of that will probably require that you get an evolution from a Psychiatrist. If that's currently clean and the history is straightforward the AME can probably issue a 1C if it's a one-time episode. Otherwise it would have to go to OKC, and that might take a while.
Look at the dispositions for the conditions in these links:
https://www.faa.gov/about/office_org...ess/exam_tech/
Also do you already hold an FAA medical? Did any of these issues exist before you got the medical? If yes, you need to talk to an aviation lawyer before doing any FAA paperwork or exercising FAA airmen privileges. If you forgot to disclose a doc visit for a head cold, they probably won't care. But OSA and mental health are hot-button issues which are associated with real-world accidents and incidents so they will go hard-over on those if they find out you didn't report it, or flew with such a condition.
Also, regarding VA disability... that's ok for bad knees, back pain, and hearing loss. But holding a rating for OSA or mental health will likely complicate your FAA medical clearance. DO NOT DO NOT DO NOT try to play the FAA and VA hands differently, they cross-check those records and they sent a Delta pilot to jail recently for doing that. If it's in your mil records or you even discussed it with the VA, better disclose it. Personally I would try to get medically cleared (legit) for those conditions and not worry about the rating. Major airline career earnings are orders of magnitude better than VA bennies. If you've already been diagnosed or documented, then just disclose it all and ride it out.