NLK,
I highly recommend you
READ THIS as it relates to controlling the inverter independantly to conserve battery power which is smart when camping without shore power.
The Tripp-Lite control panel two position switch is deactivated at the factory through a headphone style connector with wires going to the power control switch at the entry door. It does not matter what position that Tripp-Lite switch is in. Confusing, right? Why is it there? Use it only to moniter the health of the batteries.
Assuming your 2008 Tripp Lite is like my 2007, there is a little 3-position switch on the top/right area of the inverter itself, just above the reset switch. Guess what the 3-positions are? Auto Invert, OFF, and Line Charge. The control panel switch you question, is a remote extension of that control switch, but excludes the OFF position.
Unless you are trying to achieve what my "READ THIS" article accomplishes, simply ignore the position of the control panel switch as it doesn't matter. The control panel is then only useful for monitoring the condition of the batteries through the colors of the LEDs.
To answer your specific questions under the factory wiring setup...
Q: While plugged into Shore Line at the park, should the Inverter be set on Auto or Charge Only?
A: It Doesn't Matter
Q: While stored in my garage between trips, not plugged into an A/C outlet, should the Inverter be set on Auto or Charge Only?
A: It Doesn't Matter.
Q: While driving down the road, should the Inverter be set on Auto or Charge Only?
A: It Doesn't Matter
Q: In each one of these questions should the Battery on/off switch next to the steps be On or Off?
A: Killing RV power using the switch at the door kills all 12V opperations inside the RV, including the fridge. That is bad during any time you have food in the fridge. That is good during RV storage.
I editted this a couple times since intially replying. I realized I had some things wrong.