If I take AP Euro and pass the AP exam, will I be excused from the class in college?

It depends on the college, but usually not at Ivy-level schools. Not sure about the UCs.

EDIT: According to this page, UCB does not accept AP classes toward the Seven Course breadth requirement: http://ls-advise.berkeley.edu/requirement/7breadth.html

I suspect the other UCs have similar restrictions, but you will have to check the ones you are interested in.

AP Euro is definitely perceived as more rigorous than APHG, so I would take that.