If where you plan to transplant them to a spot that is ready I'd just plant more seed, but if you're out of seed or waiting for a space to become available then it might be ok to move them while still very young. Keep in mind that one beet seed can produce 2-3 sprouts, I use small scissors to snip the extra sprouts off allowing one beet/root to grow. But that being said I must confess that I've never transplanted beets, it's raining outside and this is some yadda yadda time ;-)

Beets transplant better than most root crops. I fill in blank spots in my rows with plants I've thinned from other places in the row. I've also grown early season beets from transplants that I started inside. They have done well. If you are not looking for perfectly sized beets, you can leave multiples if the groups are not too close together. Harvest, leaving one in each group, when they are baby beet size. The remaining ones will fill out in the extra space that becomes available.

As above. I do that routinely, because of irregular germination. If they're coming up a little dense in one place on the row, and there are holes in another place on the row, a teaspoon will move them to distribute better. No sweat. That way, I get evenly spaced beets. I do this with seedlings when they are tiny. One inch high or less. Water them in when done transplanting.

I thin beets when they're a few inches high - both because that's when I typically get around to it and because I LOVE baby beet greens. They are delicious. I would never transplant them just because they are too good for eating.

Once they are a few inches high, I think it is too late to effectively transplant them. If you want to transplant them, do them right after they come up. At that point, they are too small to even think of eating them.