Is it possible to jump from DIII to DI or do you have to take a job in DII first?  Are there any key prerequisites to get a DI Job?

It is possible to go directly to DI from a DIII. Others can be more specific but you'd need a couple hundred wins (I imagine) and a very successful string of NT runs in the "four year window" that has so much weight in the hiring logic

Yeah a bunch of people have done it. You need to stay at the school a very long time, rack up tons of wins, and many NT berths, even outside the 4 year window, but need deep runs IN that four year window. And you aren't going to jump to a BCS school, you'll go to a mid-major or low-major in D1.

its not entirely accurate to say you need a few hundred wins or have to stay somewhere a very long time.  i've gone D1 to D3 twice and spent 6 seasons and 5 seasons at my D3 before moving up.  In both cases I was eligible to move up after the 5th season.  I think it does require at least one run to the Elite 8 or further, as I had a similarly successful run at a D3 school (record wise) where I only made the S16 and was not eligible to move up