Yes - the inverter takes the battery voltage up to 120Vac
Your 2nd question
If the Auto Transfer Switch inside the inverter is defective then maybe the GFI could trip. It all depends on the failure mode.
My MS 300watt Sure Sine is hardwired to a ATS which is in series on the second AC breaker driving several receptacles. So if the ATS goes bad only those outlets would be affected. The inverters powers a 12Vdc dc brick which drives the ATS relay.