When updating a game that has DLC, do I need to update each piece of DLC? For example, Darkest Dungeon has patch files for the main game but not the Crimson Court expansion. When it updates, can I just use the patch or do I need to download and install the expansion again as well? This can be especially inconvenient for games that have several bits of DLC.

It probably depends on the game, but I think with the few games I have that would qualify (ToME and SOTS: The Pit are the only games I own DLC for I think) you just installed the patch and didn't need to reinstall the DLC (unless it had it's own patch).

Does the game show it's version number anywhere when you launch it? You might be able to tell what version the DLC is and if you need to do anything with it.