11 speed 1x crank with 7 speed rear cassette / shifter?
 

I have recently had an accident and my crankset is completely mangled, it is running a hollowtech 2 bottom bracket and a 7 speed rear derailleur.

Looking around my cheapest option seems to be a 1x Shimano Cues - FC-U4000 ( 1 x 40T ) for about £35 which I do know is based off the 11 speed Shimano system.

Simply put, if I am running 1x on a cheap shopping bike, would an 11 speed crankset be OK up front, with a 7 speed chain and derailleur?

Yes.

Yes. The thinner teeth are of no great importance if you are using a single chainring. They will wear slightly faster. I have run chainrings made for derailer gearing on bicycles with internal-gear hubs, and 1/8" chain. Problems would occur if you went the other way, narrow chain and wide chainring teeth. A chainwheel made for 1x actually has an advantage: tall teeth, so it is less likely to fall off.