Shadow's level, HP, MP, and Magic Power will vary. The speed stat can also vary, but the only esper that gives a stat boost upon level up to speed are the Odin and Cactuar (GBA only) magicite. Both are found in the World of Ruin, and there is no way for Shadow to level up his Speed because Shadow leaves the party after the Floating Continent and rejoins after his battle at the coliseum. The only way his Speed stat will vary is if the player used a cheating device to gain access to these magicite in the World of Balance before Shadow leaves.
Shadow has no equipment, including having no weapon equipped and will attack Unarmed. This battle is easy, but the random factor of the coliseum might make the player's character waste turns on useless spells or abilities, which would prolong the battle more than it should. Upon his defeat, Shadow will join the party.
On the Floating Continent, Shadow can receive a permanent status effect(s) like Poison, Invisible, etc. But once Ultima Weapon is defeated, Shadow leaves permanently and the player cannot heal his status effect(s). Shadow will not be found where the party has landed in the beginning of the area where Shadow is first recruited as well as leaving and returning to the continent, Shadow will not be there. If the player waits for Shadow on the Floating Continent, he will still have those status effects applied to him when the player faces him in the Dragon's Neck Coliseum.
If Shadow's opponent is Gogo and he mimics Shadow in the coliseum, Gogo will attack himself because Shadow is classed as a party member even as an opponent.