To many "ifs" in the equation to vote on it, but I'll try to answer the suggestions and throw in some new ones.
Armour weighs you down:
Not armor, but your general encumbrance. Pretty much based on buoyancy. And you stay afloat by "swimming upwards". This means that in order to stay afloat, the heavier you are the slower your forward vector becomes. A fighter will use his added strength to overcome his weight. A thief can use his added stamina for prolonged stamina effect. A mage can use his "levitation" (not as it used to be, see below) spell to thrust him upward to reduce his weight.
Can swim faster at the lost of stamina :
See above. Sure, if you don't suffer an encumbrance penalty, stamina could be traded in for speed (like sprint). But I typically see it possible for developed monks and acrobats carrying very little weight.
Bodies eventually sink:
Also, see above. Or did you mean dead bodies?
More life in water (fish):
This goes without saying.
Water weeds, plants etc:
Ditto. Algae, plankton, and corals would rock.
The deeper you go down, the darker it gets:
Nice idea, haven't thought about this. But also water should vary in clearness.
* Yellow sulfuric in volcanic tundra. Low vis.
* Brown mud in sewers. Low vis.
* Whiteish frowning water in hotsprings. Low vis.
* Whiteish river rapids. Medium vis.
* Clear mountain water. High vis.
* Blueish seawater. Medium vis.
* Blueish glacier cave water. High vis.
Where visibility also varies with weather (where appropriate) and velocity/stir.
Fire (even magic) does not work under water:
Already in. At least for torches and ranged spells. But you could add "no sitting on the seabed performing alchemy" to the list, as that is currently possible
Repairing weapons too.
Fighting under water should be overhauled:
Ehm, yeah, I guess. I don't have any ideas though. Maybe just scrap underwater fighting and do it the FONV way, like LakeLurks following you on shore and attacking you there? Anything else could be based on puzzles for big sea monsters or even fish out the smaller enemy fishes. If cold, we may have a substantial new "enemy" to worry about. If someone have means to do underwater fighting that feels natural, I'm listening.
Diving etc:
I'll have to say no on this one. Mostly for technical reasons. How do you initialize a dive? Special control combo? Complex. Jump that detects you'll land in water? How about acrobats being able to redirect their airtime? Which in real life translates to more control over where you want to land which is impossible in a game without air control.
Splashes from jumping in, falling in:
Isn't this in already? If not, then sure, very little cost. But I want the splash to vary according to the speed I "hit" the water with. If I'm stealthy and feel the water with my toe, I don't want a massive splash. In audio specially, most games get this wrong.
Armour, clothes, hair etc gets soaked:
Yes, both visually and physically. Physical wetness adds to the wet acting as a hypothermia multiplier, and it takes some time to dry up. Not needed to add weight.
Swimming in freezing water causes hypothermia:
Sort of. The location, weather, season, and time of day determines the temperature on dry land. In water, temperature and thus hypothermia points are gathered according only to location and season, but are typically more than on dry land. So after a cold bath you have gathered quite a bit of "hypothermia points" (similar to Fallout radiation). Back on land, it will continue to cool you down based on the land temperature, with decaying wetness and wind acting as multiplicators. Note that there are locations that does not pose the same threat. Hot springs should also be available to get rid of hypothermia points effectively. This is just a general outline, I haven't fully tested it in spreadsheet yet. But at worst conditions, it shouldn't really feel much different from the worst radiation conditions in Fallout. We only need a new enemy to respect and worry about, not a killer unless you keep being stupid.
Others (explain):
Repeated and/or moderate hypothermia causes disease - common flu. I mean, in how many games do you actually uncontrollably sneeze?
Could be an obstacle to sneaking. If hypothermia is part of the normal game, this could be an expansion making it to hardcoe mode.
Freeze and thaw water, especially still water and never sea water. Walking on ice should have decreased friction.
As for levitation, think of it as pointing where you want the front of a rocket engine to point and you were sitting on it. It completely replaces feather, jump, slow falling, water walking, and levitation based on your skill level. But at no point will you be able to use levitation as a means of travel. And since you need to direct your thrust, it's unsuitable for fighting. If you want to reach a ledge or house door that is hard to get to, you better prepare by making yourself light. I'm tired of mages always getting the super convenient way of everything (which is why I don't play them, they're just boring now). Constant drain, no regenerative - prepare for a consequence like everyone else.
Now, combined a little - let's have some plain old *FUN*... Water freezing in your left hand. Levitation (thruster) in your right hand. Freeze water as you hurl down the ice you create. But you already was in for a swim, so you're cold AND wet. At these speeds, the wetness and wind chill factor (due speed) now presents a much quicker gathering on hypothermia points. Probably should have waited until the wetness wore off. But of course, Skyrim being so small and all, I don't see larger real sized lakes remotely possible...