Re: No risk of signal detection on Earth from the Moon
If they want 1Gbps via 4G, they'll be using a lot of carriers, carrier aggregation and MIMO. Vanilla SISO LTE in 20MHz of spectrum (the largest a single carrier can be) offers ~75Mbps @ 64QAM.
They'll probably have it a bit easier as they won't be having large numbers of mobiles to worry about, so the RF schedulers will be quite lightweight, and the "atmospheric loss" in the propagation model will be greatly reduced (also presumably it will all mostly be line of sight, perhaps antennas built into the space suit). I don't think they'll have that much trouble streaming 4K video, if they wanted to.
Frequency plan? Probably the higher frequencies as the range is unlikely to be a worry, and no worry about getting indoor coverage.