1)You can drive or fly. If you fly you arrange your travel through carlson-wagonlit. I drove and made like 4k doing it. If you fly, you get a cab from the airport to FSH.
2) You have like 4-5 days before your report date to USUHS. If you don't go straight there you'll have to take leave. I'd definitely recommend getting your housing situation squared away before reporting to FSH. If for some reason you can't they'll put you up in Navy Lodge for something like 7-10 days (i'm not exactly sure how this works but know it can be done).
3)the one thing I'll add to dru's comments is that there is time budgeted into your orientation week for HHG delivery, ie, they give you 1-2 afternoons where you can call and have your stuff delivered.
4)You can wear any type of running shoes as long as you have socks cover your ankles which might make it a little tough if you were wanting to wear vibram five-fingers, but i've seen it done. The black skull cap can be purchased at clothing and sales. You won't use any of your winter PT gear. You can purchase that once you get to FSH or later.
The biggest piece of advice I could give to all of you right now is to just be fluid. I know you'll expect everything to be set and ready to go, but it's not. So put that type A aside and just relax and go with the flow. It will be a big cluster once you get down there. USUHS has been doing this for almost 40yrs now, and even if it seems crazy at times, they'll get you taken care of.
If you're Army start working your beret now so you're not walking around looking like G.I. Chef Boyardee