For a third class medical, the AME will usually charge $100 or less depending on your location. You can try to get insurance to cover it, but if you're not a professional, they may view it as non-mandatory.

For a cut-and-dry case of a previous and fully situational resolved situational depression, the AME may be able to call the FAA and get a verbal OK on the spot. Otherwsie it would probably take a few weeks.

Since you have an additional medical issue, I'm not sure if the FAA will need to look into that.

If you were a professional pilot (or a wannabe) I would suggest using an aviation medical consulting service BEFORE you go to the AME. However, if you just want to be a Private Pilot, it's probably cheaper to just deal with the AME even if it takes longer.

FYI, here's one consulting service: http://www.aviationmedicine.com/