This is a superior design for operating off unpaved airstrips. If the landing gear is robust enough, this could handily supplement the C130's capabilities. They can always place heaters on top to manage any ice accumulation.
there will be any number of design issues for them to solve and this will certainly be one of them. it doesn't seem insurmountable, though.
Ice usually forms on the leading edges, so if they are kept warm it will greatly reduce this risk.
No. Why would it?
this was a common problem with the older tail-mounted engines. ice would come off the wings during takeoff and go straight back into them.