All of the above are true.

A couple of other caveats:

The seller must accede to any buyer payout.

By law, they are the one paying the commission and even if a buyer was represented by a buyer agent, the seller has to approve the payout.

This can be really problematic in cases where the buyer and seller negotiations get tense or acrimonious.

Also, the rebate needs to be disclosed on the HUD statement.

If it is not, potentially the parties are acting illegally (though it may not be worth prosecuting).