Which levels count towards this?

All normal levels? Hardened levels? The new super ascend levels?

And does it keep counting after ascending? (I would assume and hope so...)

Not sure about the New SAsc levels, but don't believe Hardened counts.

All the regular level count regardless of which Ascension # you're on. So you can clear them all twice on 1 Ascension, or once each on 2, or once each across several if you skip some levels on Ascensions.

Thats great news!

Now I can safely ascend once I finished my last 3 levels (still got Hard Trisk to try, ugh 🙄).