Yes, valid US passport (and not exceeding 90 days of the last 180 days in the Schengen zone) is enough. To go back airside you need your boarding pass and your passport, and you'll have to re-clear security.

The train station is clearly signposted and typical off-peak Service is 12 trains per hour to Zurich Central Station, a mix of regional and intercity trains. The daily travel card for the local system covers them all (and trains and buses).