I ask because I'm getting back into Skyrim after a two-year absence; I don't have any DLC and I'm not sure when I would be buying them.

I would think the answer is no, and would think DLC-dependency would be avoided whenever at all possible, but I want to make sure.

Depends entirely on the mod and its author, though most kinds of mods have additional plugins for each DLC, meaning you can use it no matter what combination of them you own.

Ok, good to hear.

As stated it depends on author, but most modders want their mods available to more players. This means they try to not use assets from DLCs, since not everyone has them.

Depends on the mod & what you're looking for. For example, Dragonborn DLC re-introduces the WaterWalking enchantment, on a very limited scale (only 1 item). Mods that add WaterWalking Enchantments or Spells will likely require Dragonborn. That sort of thing. Same type of thing with Crossbows & Dawnguard.

Actually since the waterwalking enchantment is a magic effect/spell and not a mesh/content setup, it's likely that the ability to do so was hardcoded into the CK. Thus I believe non-DB mods could add waterwalking, much like how mounted combat was a 'DG' feature but added universally to the game.

You would have to check the mods description of what is required.