I've done that same train a few times this year.

Won't be an issue "NORMALLY", but at Taoyuan HSR station the bus wait to the airport may be an issue (assume 20 mins from arriving to Taoyuan HSR to get to the terminal), as you suggested a taxi to the airport is more ideal from the HSR station.

I HIGHLY suggest you do online check in.

Security clearance is quick (as it's a flight to NRT), expect 10 mins or so for security and 10 mins for immigration and DIRECT to the gate if time is short.

Her major advantage is being without checked baggage.