The card text seems to suggest they do ignore shields

"...ship is dealt one facedown damage card for each [pow] or [crit] icon rolled. If there is at least one [crit] icon, deal the first card faceup."

Doesn't mention anything about 'suffering damage' which is the terminology used in the rulebook, just about dealing out cards..

The same would seem to be true for asteroids "..ship is dealt one faceup damage card"

but not for debris which says "...suffer two damage on any one hull zone..."

It does exactly what it says. Yes it bypasses shields.

Does General Dodonna's ability trigger off of these dealt face up cards?

Does General Dodonna's ability trigger off of these dealt face up cards?

Yes, he triggers any time a face up card is dealt to the other player.