Yes they can, but it depends on the wall how well it will work. For example if there are steel studs or steel reinforcing it may not work that well. Nevco receivers for things like Locker Room Clocks are designed so that they can be mounted away from the clock itself. Plan on installing the wireless receivers so they have clear line of site to the control and if it works through walls, then that is great too.
No if you switch the battery voltage ON while the inverter is already switched to ON it will not always start up. This is the reason that the power switch on the inverter is used to start and stop the system and the reason that the controller box must be mounted in an accessible location.