Are you, as DM, targetting characters' saving throws they are not proficient in or do you let the combat play out naturally.
That depends. Experienced NPCs whose lives depend on it might have acquired knowledge of some saves. For a character looks like a martial they might guess that effects targeting their Wisdom or Intelligence will be more successful (sadly, in D&D stereotypes are backed up by mechanics). For a dwarf they might guess that poison isn't the best damage type. Automatons just carry out their orders, no knowledge applied. I do assume that natural selection in a magical world has lead to natural creatures - animals - having reasonable survival instincts. So knowing to attack a weak stat isn't purely a matter of sentience.
Are you, as DM, delibertately targetting characters with fewer hit points, or do you let the combat play out naturally.
That depends. Right now in OOTA a Shadow Demon is alone with a character it has just taken down. It is going to spend an action to deprive that character of two death saves. Another NPC - a Medusa - is going to loose arrows at the character it has already wounded, hoping to kill them. Earlier, stone soldiers simply battered whichever character was nearest (dog-piling onto the hapless bard!)
There must also be an element of good-for-the-gander etc, if I allow my PCs to act based on knowing that a foe is hurt, likely to be weak to certain spells or whatever, then that is because those things are able to be known and perceived in the background world.
Given that spells can be combat changers, I don't agree that the casting of spells is something minor.
Whether it is or is not, it is simply one of the many things a DM must adjudicate with a sense of fairness, drama and good humour. I like to surprise my players, and am delighted when they surprise me.
Sure, given the nature of counterspelling where you directly negate a pc's action I think it is quite different to the examples you've listed above, where DM knowledge plays an active in-your-face role.
It seems really the same to me. Player does X, I arbitrate what happens. Could be a Counterspell, could be an attack intended to break Concentration, could be a lot of things. It's all part of the craft of DMing.
Are you saying that you expect DM's to be bad at exercising objectivity and fairness in choosing NPC actions? Are you saying that even if the NPC doesn't make their check, you expect that the DM is going to be incapable of playing that out appropriately? Is that right?
So you roll for every spell the PCs cast or only the spells that you would like to counterspell?
Rolling can be fun, personally in this instance I don't believe it is necessary or fun - rather implement a passive arcana check to see if the caster recognised the spell. Fair for all and faster at the table.
Back up the thread I think we reached a point where the proposal was
1. Passive Arcana or Religion DC 12+ spell level (or should this be slot level) to know the name
2. Active Arcana or Religion Dc 12 + spell level (or should this be spell level?) to know the level of spell slot it was cast with and any metamagic
Intelligence (Arcana) for arcane spells
Intelligence (Religion) for divine spells