I read in a couple places that, unlike other weapons with a fixed base damage, Skyforged Steel weapons actually scale in their base damage as you level. If that's inaccurate, feel free to stop me now. If it is accurate, though...

If you enchant a Skyforged Steel weapon with fire damage or something, does it keep scaling its damage as you level, or is it now no longer considered "Skyforged Steel" and stay fixed like every other weapon?

All weapons "scale" as you level up the associated skill that governs the weapon.

Damage rating displayed when you look at a weapon isn't just the weapon's base damage. It's the weapon base damage + the modifier due to your skill in using that type of weapon, hence the "scaling".

Weapons don't scale in the sense you're getting at.

A weapon has certain levels, such as 26 or something. If the player is level 26, they will get the level 26 weapon. If they were anywhere between 16 and 26, they would get the level 16 weapon. That continues for some time. Each level has more and more damage, but the weapons are always fixed at the level they were acquired at.