Excellent questions - you're doing a good thought process and that will help a lot. I went through a similar decision, though wanted an open trailer for easier storage.

Got an F150. They have good reputations for a reason. Got a 2000 with Xtracab so can carry four adults locally (my tow vehicle budget was $9K).
Considered Chevy but model naming/numbering is confusing, and Tundras were 30% costlier.

For your very limited use, you should have no problem with the $4K range.
I'd have a van but wanted 4x4 for snow/camping. Vans seem most efficient - one big box. Especially for a small/open trailer. Put fuel and smelly stuff on the trailer and everything else in the van. Camp in it at track or in the woods, change clothes comfortably, or carry seven folks in comfort.

To me, SUVs have the least-space efficiency with the worst aspects of both trucks and vans. On the other hand, there are a LOT out there.

You need bigger than a Ranger/Explorer, since a closed trailer and Miata will be 7-8,000 pounds fully loaded. A mid-90s F250 might be perfect - as pickup, van (E250), or Expedition.
Others can advise much more on trailers - the issue is not what you can make go forward, but how much weight you can stop.

Hope this helps - it's my first post here after lurking a month!