I moved from West Virginia to the Bay area of San Francisco about a year and a half ago. My situation was different from yours, since I relocated with my company, but I was hired with some non-company experience hires as well, so I'll include their experience.
I don't necessarily see Chemical Engineering as a high-dollar position (maybe low-high, but not high), but I know that it is treated very differently than something like a teacher or nurse which have much larger hiring pools.
-Companies don't pay transportation for interviews unless it's a high-dollar/high-profile position.
My interview trip was paid for, as were trips for others coming from outside the company
-Companies wouldn't pay for relocation(again, unless it's a high-dollar position).
Mine was completely paid for, and those experience outside hires also had a package, but it was less generous.
-Most companies only look at non-local candidates if there aren't any suitable local ones.
For my company, this was interesting. They try as hard as they can to hire within the company locally, then within the company nationwide, then the location of those outside the company doesn't really matter.
-Apartment rentals might be hard to get without already having a job history in the area.
This really depends on the location and the landlord. Mine was happy with paystubs from WV and a credit report.
I went apartment hunting about 3 weeks before I moved to the area; the "house hunting" trip was paid for by the company, but far cheaper than staying in a hotel until the place was ready. This was almost too long before the move. As it was, the landlord made me pay for 4 days that I wouldn't be there, since he would be ready for me then. If I had fought it I may have gotten him to not require it, but I didn't want to lose the place.
My advice would be to start applying for jobs as soon as you're able to leave, but don't quit your current job before getting another. You can feel out different areas that have opportunities and not have to worry about being unemployed. If there's difficulty in finding someone that will interview you without meeting you, I'd try to suggest a Skype interview as a first step, and see if they are serious. Don't sell yourself short. Look for areas that have large technical support companies, and figure out if you would/could live near them, apply for any opening and see what happens.
Good luck!