There generally are direct trains but due to the freight train derailment in the Gotthard Tunnel international EC trains all run to/from Zurich.

The easiest way to do this route is to take the hourly RE to Lugano and change there to any train to Arth-Goldau/Lucerne. Don't worry about the 4 min connection in Lugano. Fully reservation-free.

To answer all your other questions use a reliable planner like sbb.ch, bahn.com, SBB app or DB Navigator app.