Are there special dice associated with these cards?
Yes they have custom dice
Are they treated like character cards in that my opponent can't buy the dice associated with them?
Yes they are
Do they have a dice limit like characters?
Yes they do
I didn't see any images of Non-basic Action cards in the rules, so I'm having a hard time figuring out how they are used.
As far as i know there are only 2 in the AvX set. The common version of these two(Mjolnir and Vibranium Shield) can be seen in this image.