Does the Denver airport have a phone one can use to call Microtel Inn for the shuttle? Or, does the Microtel shuttle run routinely?

Routinely. But you can also call. I think it's every 20 minutes

The shuttle runs at certain times and yes contact the hotel to provide more efficient service when shuttle is needed. The shuttle is much less of a hassle with a phone call to the Microtel Inn.

There's a train stop (from either the airport or Denver) that's close to the hotel and I'm almost positive the hotel has a shuttle that can pick you up and drop you off.