You can access Sephiroth's equip menu during the Kalm flashback, you just can't mess with his stuff. You can't control him during battles, but let's face it, you don't really need to.

I think his walking through the flame wasn't because he was immune to fire. He just watched where he stepped.

Why he's invulnerable to monsters and stuff is obvious from gameplay perspective (people like me wander around indefinately with him, so sooner or later he'd wear down, therefore he must be invincible). But as for logic... that I don't know.