Probably not?
D&D 5e breaks from the tradition of several more recent editions of D&D by allowing - indeed, encouraging - DMs to make their own rulings on situations rather than following a single explicit canon in regards to the rules. It also - particularly in regards to spells - attempts to be descriptive (of the general effects and feeling of the spell) rather than prescriptive (giving an exact, itemised letter-of-the-law list of the spell's results) and so leaves many details of precise interpretation up to individual rulings.
The sleep spell, in this instance, lists:
falls unconscious until the spell ends, the sleeper takes damage, or
someone uses an action to shake or slap the sleeper awake.
A purely prescriptive reading would suggest that this means exactly what it says - the sleep spell causes the target to fall unconscious until one of those three precise things occur. It doesn't matter if someone tries to wake the sleeper up by pulling on their hair or putting a wet cloth over their face; it doesn't matter if there's an earthquake or someone blows a trumpet in their ear. They wake up when shaken or slapped (by someone spending an action, not by any natural phenomena), the spell ends or they take damage.
A reading more in line with the spirit of the rules, on the other hand, suggests that sleep sends you into a deep magical slumber - but one that can be awoken from. What exactly it is that wakes a person from the slumber is up to the DM - but something on the same general level of violence as slapped or shaken would do. I personally would probably count falling into cold water to be enough. But maybe not if, say, the target is a merfolk or something.
This is absolutely a decision that should probably be made as it comes up in game. Maybe, as suggested by bacrossland's answer, after the spell has just been cast, with the conceit that the character didn't know whether or not water would be enough to wake the target - but be aware that this could seem unfair to a player who expected that their character would know whether the spell would work or not.
Cold or warm water?
Is cold water enough to wake someone up in general? Yes, assuming they're capable of being woken up (e.g, they're not exhausted because they've been swimming for days). But warm water may not always be as effective. The key thing to consider here is the mammalian diving reflex - it causes several changes to metabolism, breathing, and almost certainly wakes the sleeping, and it triggers from cold water on the face region. This is why splashing your face with cold water in the morning not only wakes you up, but makes you feel refreshed. It doesn't apply to warm water. Again, DM's call based on a general understanding of biology / decision about magical slumber as to whether Randy would be woken up in the warm or the cool section of the Roman bath - just giving you some background biology information here.
RAW, they will not awaken.
The sleep spell clearly lists the ways to awaken the sleeper. This is magical slumber, which can occur even in the pitch of mortal combat (in fact usually in the pitch of mortal combat). It is not ordinary sleep. Note that the action to awaken the sleeper is to take damage, or be shaken or slapped. The spell says nothing about loud noises, bright lights or other things that normally awaken sleepers.
The entangle spell has no language in it that describes a level of violence even so mild as a slap or shake. A DM is free to interpret this otherwise (that's their job), but there is no specific language in the rules to suggest that the entangle spell will awaken a magically slept person.
Best Answer
The damage resolves on a per attack basis
In the PHB between pages 193 and 194 is "Making an Attack" header, which specifies the order of an attack. Resolving the attack is when you deal damage.
For instance, with Extra Attack at level 5, you're making 2 attack rolls and therefore 2 instances of damage. If they're unconscious from sleep, the first attack automatically crits if you hit their AC, then they wake up, and the second attack does not automatically crit. It's important to note that you only get automatic crits on an unconscious target if you are within 5 feet of it, and the same applies for advantage on prone targets.
The same logic applies for spells, any time you make a spell attack roll, you're doing an instance of damage. Spells that add to your attack damage, like smite spells, add on to the initial damage, so that damage is a crit as well for the first attack.
For things like Magic Missile, I've written an answer on another question that may be helpful.
For spells without attack rolls, you need to look at it on a case by case basis. I can't think of a spell besides magic missile that would appear to have multiple instances of damage within itself that are not separated by time or triggers.
The same logic applies as above: if you're making separate attack rolls, you're making separate attacks. In addition, bonus actions and actions happen at different points in time. See page 189 of the PHB
If you were granted an attack or spell with a bonus action that dealt damage, it would crit (if within 5 feet of the sleeping target), and any actions after that would not automatically crit. Once again, smite spells like Thunderous Smite are cast as a bonus action, but only trigger damage once you hit with a melee weapon and their damage is added, which makes it a part of your melee attack damage. The first attack with smite will crit, so double all of your damage dice, including the smite dice.
Side note Because of questions like these, I highly recommend rolling your attacks and damage separately. Some players are tempted to roll all attacks together and then roll their damage all at once for ease of use (or sometimes roll damage once and double/triple the damage! The horror!), and cutting corners like that can, in this case, miscalculate critical damage or deal too much damage to one person (you can split your attacks up between movement!).