Starcraft – Does Archon Mode count for any Multiplayer achievements

It does seem so.
I play random and got an achievement for 50 wins as random as well as the 10 wins as terran, zerg and protoss achievements.

Reposted from comments, as suggested above:

Well, the thing about the F9 achievement screen is that it should open up the "group of missions" that includes the mission you're currently on, and additionally separates achieved from unachieved. Scroll down and you should find all achievements for that mission in a row (assuming you haven't completed any before).

What I liked to do (when I was completing SC2 achievements) was view the achievements before the mission, when you're on the Hyperion, so that I would know what to strive for immediately, going in.