Anyone have any thoughts on pros/cons of touring on 650c wheels and tires? I understand that they may be more difficult to find in the middle of nowhere, so enough spares would be brought with from the beginning (in either 650x23 or 650x28 (that Terry makes)). I would be using a "BOB"-style trailer so not as much weight would be directly on the wheels. Any thoughts?

What style/duration of tour? For a 2 week tour they should be fine but a year-long continental tour would be pushing your luck regarding spares.
Assume that you wont be able to get touring tyres anywhere (pretty much the situation with 700c in many parts of the world)
Take at least one spare 28mm tyre and have backup spares ready to be sent out to you.
If you are considering a bike for purchase, use 26" MTB wheels rather than 650c.

It would be a few months, but only within the US. I would bring a healthy supply of tires and tubes. As I am not the tallest, I think the smaller size would be a better fit on longer trips. And I like to go a bit faster than I think 26" (even if they are slicks) would allow. Any recommendations about pannieers or trailers for 650c wheels?

There is little practical difference between a 1" slick MTb tyre and a 28mm 650c touring tyre. You are perhaps trying to compare a racing setup with an off-road one. If you have the choice go for 26" MTB.
An MTB wheel does not have to mean an MTB bike; it is just another wheel size. There are lightweight touring bikes built for MTB wheels.

26" wheels are inherently stronger than 700C. Smaller diameter wheels are always stronger than larger diameter wheels. That's one of the reasons that 26" are used in mountain bikes. If you are going to tour in the US, I'd suggest 26" rather than 650C. The rim diameter of the 26" is 559mm while the rim diameter of a 650C is 571mm. The size difference won't make that much difference in speed but finding 650C tires outside of a major metro area would be difficult. 26" tires are going to be as easily found as finding the nearest Walmart - but only if you have to You want to save you bag space for stuff other than tires and tubes.

All other things being equal, there should be no problems with bags or trailers on 650C or 26" wheel bikes. Lots of mountain bikes have been used for touring.

Edit: Forgot to mention wheels or rims. Those are going to be extremely hard to find even in major metro areas.