Can I refreeze beef mince?

Because you let it thaw at a relatively high temperature, (ideally it should have been thawed in a fridge), I'd say it's best you cook the burgers and then freeze them. I wouldn't refreeze your raw mince.

The official answer is, no, it's not safe to refreeze. But the common sense answer is that, if the mince was not defrosted in a high temperature environment, or microwaved and was quickly made into burgers and refrozen, you are probably in no danger.

cook them 3/4 done!! that way wehn you want them they will finish cooking and heating all at the same time. Refreezing meats eliminates some of the quality of the meat! I usually cook all of a package of ground/minced meat, then freeze what I don't want for that meal.

Yes of course. Freeze and refreeze things as often as you like. Each time they will get more and more mushy and unhealthy but its your beef so do what you feel works for you.

It's only safe if beef still has ice crystals.

not unless you've cooked it, like in a spaghetti sauce or like sloppy joe.

Thawing the meat at room temp is the real problem.