#1 - If on a 3rd strike the batter reaches 1st because the pitcher threw a wild pitch, is the pitcher given an error?

#2 - If on a 3rd strike the batter reaches 1st because the catcher allow a pass ball, is the catcher given an error?

No to both. Score it as K+WP or K+PB.

Third strike, dropped ball
 

If...
K+WP and K+PB is the way to score these plays, then is it correct to score a third strike, dropped ball a K+DB? The official score keepers at my games have asked how to score the dropped ball - third strike. My reply is, "I don't know. If it isn't in the rule book, I don't have the answer." Let's hear from some of you 'official scorekeepers'. Is there a different designation when B1 makes first base and when he doesn't?

If the batter/runner is thrown out before reaching base, score it 2-3, or however the out is made. If the catcher tags the runner, score it 2-U.

If it's a DROPPED third strike, record it as K+PB. If it's a third strike NOT CAUGHT, determine if it's a PB or WP.

Bob