On my Imperial, I would like to get a steel sword, upgrade it to "Fine" and enchant/rename it and carry it throughout my game. But, if I upgrade to "Fine" now, can I upgrade again to "Superior", and so on after getting the perk that lets you upgrade enchanted weapons?

Thanks!

Yes

Yes, you can re-smith weapons, if your smithing ability has improved. But it's better to do it only once, since you need an ingot for each smithing job. Not a big deal with iron, but ebony or glass is another matter.

Absolutely.

As long as you work on your Smithing Skill, as your score goes higher, so does the quality of the weapon you smith. So you can take that same Steel Sword all the way up to Legendary as long as you have the appropriate Perks. Even then, if you use several of the Blacksmith's Philters, Potions, Elixers, etc., then your weapon will go even higher in damage base.

Alright, thanks! I said steel up there, but I meant Imperial. I just thought that there would be some roleplay value to carrying a sword I started out with, maybe brought with me from Cyrrodiil.

As in, it was taken from me when captured, but I got it back.

You definitely can keep upgrading, but you could also enchant the sword. That would help improve it even if your smithing is low. Just be careful that you put a high-level enchantment on, because you can't change your mind later.

Probably Soul Trap or something of the sort, I don't really want to enchant it, but its the only way to rename it.