Is there a standard tripod mount? Is it reasonable to assume that a given tripod will attach successfully to a given camera, or will I have to research my options?
Yes. The tripod thread is standard: 1/4-20, which means ¼", with 20 threads per inch.
This is specified by ISO 1222:2010. I'm not willing to pay the $57 for my own copy, but I'm kind of curious, as wikipedia says that the current standard also allows 3/8-16 — apparently that's an older mostly-European standard. This is probably old-hat to aficionados of classic field cameras, but was new to me — all of the modern Japanese DSLRs and compacts I've seen use the 1/4-20 thread. 3/8-16 may still be common for larger-format cameras. I think it's safe to say that for consumer and mass-market professional cameras, 1/4-20 is now almost universal.
The other important thing is that almost all modern tripods use a quick-release plate system. There's a small plate which has the tripod thread which screws directly to the camera, and then that snaps into the tripod head using a proprietary-to-each-company attachment. That means that even if your camera would use the less-common thread, you could get a plate that matches — for example, a Gitzo plate which comes with both threads. And adapters between these two threadings are readily available.