You could try soaking rags in Jeyes Fluid and tying them to fence wires/posts. Other than that I have no idea.

Not legally as they are protected. Turf full of worms and sweet corn attract them. You may be better off finding ways of not attracting them rather than repelling them.

On our allotments the local badgers aren't much of a pest other than going mad for sweet corn. But then some of the neighbours feed them so they aren't that hungry.

Badgers are notorious for keeping to the paths they always use. I kept them out for a couple of years by sprinkling a lot of chilli pepper where they came into the garden and around the things they were digging up, but this winter they have come back. Wouldn't mind so much if we ever saw them...just the odd paw print here and there.
But see Roythefoots's reply to the question Several Large Holes about six questions below this one...