Yes and yes, you can use solar on the truck and power the house battery system in the trailer.
BUT you'll need a bit of a beefed up wiring system!
Think of it and design it like a portable system, which in a way it is.
You might be better off putting the batteries and everything else right near the trailer and running the power cords between the two vehicles. That would minimize the size of power cords you need.
You're always better off using 12V instead of 120 but if you have to you have to. Get the smallest inverter (or more than 1) that will power the lowest wattage 120 appliances you can. I have a 400W inverter hard wired to a couple of 120 outlets but it's got a switch built in that I can reach from the living quarters that shuts it down if not in use. I also have a 100 watt portable that plugs in to a marinco 12v outlet that sees a whole lot more use than the big one does.
Just don't drive off with the cord still attached and the trailer parked....
Don't laugh it's been done!