Up to the DM, but RAW no
Does it mean the creature still benefits from its dexterity, both in terms of mechanics and in-game world?
An unconscious creature does not use its dexterity narratively, but in terms of mechanics its AC does not change. This discrepancy was explicitly taken into account in previous versions. For instance, 3.x version had three ...
You would not fail a death saving throw from taking damage while the ward is up.
The rules for Damage at 0 Hit Points say (PHB, pg. 197):
If you take any damage while you have 0 hit points, you suffer a death saving throw failure.
If you have your Arcane Ward (PHB, pg. 115) up while unconscious and are attacked, then:
Whenever you take damage, the ward ...
What happens to their ward:
Does it drop to 0 hit points instead of the wizard? No, since it's not a damage dealing effect, the ward is not affected.
Does it disappear when the wizard drops to 0 hit points? No: there's nothing in its description that would make it do this. The wizard is at 0, but their ward is still as effective as it were if said wizard ...
An unconscious creature is not affected.
The banshee's Wail action can only affect creatures "that can hear her":
All other creatures within 30 feet of her that can hear her must make a DC 13 Constitution saving throw.
However, most notably, part of the description of the unconscious condition states:
An unconscious creature is incapacitated, ...
The answer provided states that RAW but you may want to clarify with your DM.
Personally regarding spells that remain in play like this as a DM I take it on a spell by spell basis and what makes most sense if a player falls unconscious or dies and then communicate that to my players up front. But I have played at tables where the DM explicitly state’s that ...
Disguise Self does not end when you are incapacitated
According to the rules on spell duration in the Player's Handbook
A spell's duration is the length of time the spell persists. A duration can be expressed in rounds, minutes, hours, or even years. Some spells specify that their effects last until the spells are dispelled or destroyed.
There is no ...