I've done an add-on doing pretty much that for a client. It can translate into multiple languages, and can also optionally translate node names, DM's, and NF tickets and FAQ's. Yes it works with the Google API and can translate all existing posts and new ones as they are submitted/edited.
Depending on the volume of data you have, the API can get costly (it's just Google's pricing). It's best you study their pricing carefully and do a rough calculation of how much of an initial cost (for existing posts) plus cost per month (for upcoming posts) you'd be expecting. Translations are cached locally (with an optional expiry time) once they are already made.