But they are trying to kill you. Unless the enemy walked into a room and they were already down they were just trying to kill you and are almost certainly trying to get back in the fight and try to kill you again unless you can do something about it.
We're talking about someone who is incapacitated and dying, if not already dead.
I wouldn't waste a stab on them if I had a couple of other people hitting me, unless I had a really good reason to expect they to become a threat again immediately.
(Of course as a DM, I may know that they are still making death saves, and they have a character with Healing Word in the party, and so if I actually want to kill a character I should attack them while they're making death saves. But that is not the viewpoint of most of the opponents that the DM plays.)
The only time people actively stabbing you would have priority was if you were metagaming it and know you could take one of them down this turn. If you can't do that this turn, before the downed enemy gets back in the fight, you will take less damage by killing the enemy that is down. If you can down another this turn then sure, go for a standing enemy, but in most cases, it would be metagaming to know that.
No, I would regard most creatures as valuing their own life, and probably not aware of the party's capabilities until demonstrated. Expecting a dying creature to do anything other than bleed out or stabilise and be available to be killed at your leisure when the fight is over seems to be logical for most opponents the party face. Unless they had reason to expect them to be getting back up, I'd regard making that assumption rather "metagamey" myself.
I would try to end a threat, putting someone down does not do that. Killing them does. This happens IRL in "melee" too where someoene gets hit with a bat or stambbed and then the killer hits the guy again while he is on the ground or stabs him a bunch more times.
Really? Do you have any solid examples of this happening often?
Hitting someone who is already on the ground is likely if they are still fighting back, or if they were the main threat/target of violence. Hitting an unconscious person and ignoring your other opponents who are currently hitting you is behaviour I would find unusual in a general combat situation.
They are added on as legal familiars in WOTC campaigns. The Tressym has 5hps I think, which is still very vulnerable, but 4 more hps than an owl. Tressym has a chance with a good save to survive a few low damage AOEs and with a lucky roll might survive a single melee attack from a very low damage enemy. Gelatinous ice cube has I think 18hps, which makes it the tank of familiars.
Isn't that specifically a special reward for DMs who ran games at specific event(s)?
Not something that is common even in AL, and likely mostly unheard of outside it.