Do I need to install Yuri’s Revenge and Red Alert 2 to use the Apocalypse mod

You will only be able to play (even enter the game lobbies) or people with exactly the same game version as you. The normal servers will work fine for you. I suggest you search the mods' own forums for other users to try and arrange a game.

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.