I'd expect the College to have literally everything magical I could ever need... but I cna't seem to find the arcane enchanter or the alchemy lab. Where are they?

In the hall where most of the teachers live, not the one they give you, it's on the second lvl. The name is "hall of ...something that starts with a c " there is both a enchanter and a alchemy lab

The Hall of Countenance, the Archmage's quarters, and I believe the Jarl's longhouse also has one.

Interestingly and somewhat annoyingly, there's no blacksmithing area anywhere in Winterhold.

^ Mages don't need their daggers sharpened obviously.