Steam – Would switching all the mods from the Steam Workshop to Nexus affect the current saves

Concerning the TR_XXX.esm, you're most likely trying to use Tamriel Rebuilt (good choice!) Antediluvian Secrets, which requires Map 1 (Telvannis) as well, so I guess the latter has been forgotten.

Concerning the warning, this means that one of the plugins you are using was saved using a different version of the game main files, e.g. native Morrowind.esm while you're using the Morrowind Patch Project. You can usually ignore this, although you should at least once check whether you're accidentally using a Morrowind-without-Bloodmoon version of a plugin that also has a Morrowind-with-Bloodmoon version if you have Bloodmoon installed or vice vers.

If you want to get rid of the warning for good, you have to open the respective plugin's .esp file with the TES Construction Set (using your actual Morrowind.esm etc versions as base files) and simply save it again (keep a backup though). That should update the claimed dependency without modifying anything else.

The problem you'll face are the same that what you'll have if you mix mods from the same source: some mods may be incompatible.

The only thing I can see that may be specific to the workshop is that workshop mods can take the priority, so if you have two mods that, for example, changes the same texture, the workshop mod may force its texture when updating itself.