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1) Usually you'll get an auto-enhanced score pulled.
2) You're stuck, you cannot find out this score unless you apply for an autoloan at a dealership typically.
3) Lots of places, YMMV re: rate. Auto loans are secured against the vehicle, this isn't a difficult loan to obtain for nearly anyone.
4) You're not likely gold-plated enough to qualify for some / all of the well-qualified-buyer plans, but check around. Some are more lenient than others (Hyundai / Toyota vs. Honda from reports here). The Auto Loan Approvals thread is a gold-mine for lender research.
5) CU's can be very good APR wise, and they're more apt to reward loyalty.
6) Yeah, you're sort of going to look like a "first time buyer;" however, you already have a mortgage which will go a non-trivial way I suspect for an auto-enhanced pull.

I think you are in a good shape with scores above 700 and (good0standing)mortgage.