http://iscoresports.com/

Runner on 3rd
Catcher attempts a back pick at the runner on 3rd
Throw hits runner
Ball is out of reach of 3rd baseman
Runner scores.

This isn't an earned run or is it?

Who is the error charged to if it is in fact an error?

From your description, this should be an error charged to the catcher fro the throw.

Earned versus unearned is more difficult to determine without other information on what happened during the rest of the inning.