As far as suppliers go, I can't help you. I get all my styrene from my local hobby shop. If you want to mail-order from them, I can PM you the name and phone number.
For what thickness to use, it comes down to ease of use/strength vs. scale appearance. If it's a street rod with boxed frame rails, I would say go with solid strip. It will be strongest and it is consistently sized for the whole length. If you cut frame rails from sheet, it's very easy to have one end turn out taller than the other.
If the frame rails will not be boxed, I'd say glue up a C-section at the exposed ends of the rails only, like the front of a '32 Ford. Let the hidden parts remain solid (boxed). If you don't turn it over or put it on jack stands with mirrors under it, your secret is safe. That's a trick I learned from Gerald Wingrove's books.
For glue, I just love Weld-On 3. It makes a weld joint and dries fairly quickly. I used Testors liquid cement for years until someone told me that Weld-On dries so much faster. Testors can remain tacky after days or even weeks when doing things like laminating plastic.
For filler, I like 2-part catalyzed putty. My favorite is Evercoat Eurosoft glazing putty. It sands and feathers beautifully. You may find it sticks better to primer than bare plastic.
YMMV