I GOT JOKES AND JOKES... ok Ill be nice :p
At RTC you will play along with the rest of the kids until about your 6th week in. You will then do whats called "special physicals". Aviation, sub, seals, swic, and nukes (others too Im sure, been a while). You will do the some degrading physical stuff... When I say head to toe, I mean head to toe.
As far as the peeling hands and feet... Cry about it as soon as you get there. If its a legitimate condition, try to get prescription lotion before you go. They will cater to that. You also have a tiny lock box you can keep random stuff in.
As far as the condition goes, as long it doesn't require steroids, and doesnt interfere with your ability to climb ladders and don a facemask, you should be ok.