1. Only after you have recieved your type-rating may you log those legs, and yes they must be Part 91

2. You could log it as dual/PIC. Since you probably need experience with the aircraft. Now once again, you must already have the Type-Rating in this case. Otherwise just straight dual-given.

3. The company ops specs under 135 must require an SIC, or they have an approved SIC program and it's in writing for you to log it as such. I would log it as total time personally, not SIC

4. As long as you feel you're winning here. Is there anyway you can get it all in writing? Are you getting the type-rating at the beggining, or after a year? I know it's "intern," but there's another thread about someone being an "intern" and basically being taken advantage of. My understanding is at least that this type is being paid by the company, not you. Are they at least covering your expenses on the road?