Quoting basic rules because my PH is not here.
As others have said, an attack against an Unconscious or Paralyzed creature has advantage and is a critical if within 5 feet. Against a restrained opponent (e.g. tied up), they just have advantage.
This can cause death under the Instant Death rule (p. 76).
Massive damage can kill you instantly. When damage reduces you to 0
hit points and there is damage remaining, you die if the remaining
damage equals or exceeds your hit point maximum.
Against an Unconscious Opponent (p. 76).
Damage at 0 Hit Points. If you take any damage while you have 0 hit
points, you suffer a death saving throw failure. If the damage is
from a critical hit, you suffer two failures instead. If the damage
equals or exceeds your hit point maximum, you suffer instant death.
So 1 adjacent opponent will cause 2 failed death saving throws - one more and you're gone (which could be the one you have to make on your turn).
Considering your sleep spell question - under RAW there is no way to dispatch a sleeping opponent without risking their waking. You can attack with advantage and get a critical hit - this may render them unconscious (or kill them outright) but if it doesn't then they will wake up. Given the description of the spell this seems fair enough, getting into a position where you can slit their throat is likely to jostle them enough to wake them before you do so.
Remember, all this occurs during combat conditions - you can't concentrate exclusively on the helpless creature and still look out for danger.
Parenthetically, this is normally only an issue for unconscious PCs, unconscious monsters are normally dead. A fair enough rule in my opinion as going around the battlefield dispatching unconscious opponents provides little role-play opportunity, doesn't advance the story and is a little bit ghoulish.