Which versions of Final Fantasy VI include multiplayer battle support

From Wikipedia

Final Fantasy VI's PlayStation re-release is very similar to the original Japanese release as seen on the Super Famicom. With the exception of the addition of two full motion video opening and ending sequences and new effects used for the start and end of battles, the graphics, music and sound are left unchanged from the original version. The only notable changes to gameplay (in addition to loading times not present in the cartridge versions) involve the correction of a few software bugs from the original and the addition of a new "memo save" feature, allowing players to quickly save their progress to the PlayStation's RAM.

(Emphasis mine)

So to summarize, the "game changing" differences are:

• PS version has loading time which weren't present on the SNES version.
• PS version has some bug fixes.
• PS version add a "quick save" feature (I assume it means making a temporary save to the RAM rather than to the memory card).

The only reason to keep your heroes low level until the end of the game would be espers: specific espers grant stat bonuses on level up, and the best ones are late game. This is by no means necessary and I don't recommend doing it unless you're really trying to min-max.

You should, however, attempt to have most characters at a decent level by the end of the game (40ish), as you'll need to use 12 characters for the final dungeon.

Note that characters do not gain stats on their own, they only game them from espers at level up.