I would use #4 for sure. 600w (if you ever used that many at once--not likely) would pull 60 amps from the battery bank, and #4 is good for 100 amps.
Some inverter manuals say you need fat enough wire to also to take their surge limits, usually twice the regular limit. So they might be fussing about 1200w and 120 amps (but the 80a fuse doesn't match that)
No idea about breakers etc.
No, you don't need to "ground" the inverter, but it could have a "chassis ground lug" on it. that takes a wire (#8 usually specified) to go to the trailer's frame. That is supposed to reduce RF interference with the TV showing lines across the picture, or ruining your neighbour's ham radio work. Otherwise the inverter only needs a neg and pos wire to the battery pos and neg posts and the battery does not need a neg ground wire if all it does is power the inverter.
Most times the battery is also powering other things in the rig which have the frame as their neg paths, so the battery must be grounded to the frame to complete those neg paths. But the inverter doesn't need that. It has the chassis ground just for RF reduction.
NEVER bring the chassis ground over to the inverter's Neg input terminal. It must go to the frame separately.