First off, don't forget that there's a Sleep spell in the main book that uses Thaumaturgy (YS 293), which would make a good baseline.

Apart from that, it looks about right. I'm not sure you'd want to have the Aspect be "Terribly Sleepy", since invoking that wouldn't actually put them to sleep. Invoking an Aspect that implies something will happen when invoked (such as say, "Off Balance") doesn't make that happen; it just makes it easier for someone to make it happen. Invoking "Off-Balance" won't automatically knock someone over, you'd still have to roll Fists or Athletics or whatever to actually knock them over; invoking the Aspect would get you the +2/reroll making it much more likely.

It might be better to have the Aspect be "Light Sleeper" or "Catnapping", since that puts them out straight out (which is what you you want), but still lets people invoke to get a bonus to sneaking or whatnot, but at the same time implies that the target can be woken up easily.

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.