I see [No-Intro] sets for the Master System and Game Gear, but I don't see one for Sega Cards. [...] were these games ever dumped? If so, did No-Intro just roll them into the Master System ROM set? And if they did that, is there a way to tell them apart via internal ROM header information? I don't like lumping the cards and cartridges together like they're the same thing, even if they map the same way on the bus.
All of the retail card games have been dumped - doing so is more difficult than dumping cartridges, but far from impossible. AFAIK no-one distinguishes between dumps of cards vs cartridges - as AWJ says, most games which were originally available on card were re-released on cartridges containing exactly the same data. For those games, there is no way to tell from the .sms file whether it came from a card or cartridge. For most SMS games, of course, this is not an issue - they were only available on cartridge. There are also a few games (at least prototypes, if not final versions) which were only produced on card.
what happens if you have a Sega Card and a Master System cartridge inserted at the same time? I presume it'll choose one and ignore the other, but not sure which has priority.
The (original) Master System can access ROM from four physical locations - the internal BIOS, card slot, cartridge slot and the rear expansion port. One of these at a time must be mapped into addresses $0000 - $bfff. Which one is mapped is controlled using I/O port $3e
When the console boots up, the BIOS is mapped into these addresses - it copies some code to RAM which enables one slot at a time (in the order card, cartridge, expansion) and executes code from the first of these which is non-empty. How the BIOS detects a slot as non-empty differs from version to version, because BIOSes in some regions have to support running software with no internal header.