Let's say I have a Daedric Helmet. I enchant this Helmet, play a few levels, then I realize I want to re-enchant this helmet. Could I do that?

What if I wanted to keep that helmet AND the enchantment, but now I just want it stronger?

See, the Imperial Helmet (Closed) is very rare, only 3 existing in the whole game, so I don't want to 'waste' them.

Also, any way to completely stop my character from being caught on fire? I hate when I hold up my shield and try to defend against a fire attack only to be caught on fire anyway, really ruins the immersion.

No you cannot re-enchant something, nor can you strengthen the enchantment that is already on it.

As for the being on fire thing, no there is not a way to prevent this.

to the first question, no, not without a mod.
to the second question, spellbreaker?

You can strengthen the armor rating of your enchanted gear with smithing and the arcane enchanter perk.

You can not strengthen an enchantment without re-enchanting another piece of equipment.

You can stop your character from catching on fire with Spellbreaker(a shield),elemental protection(a shield perk that blocks 50% of magick),or a ward spell,( that blocks all damage until the spell is broken).