You have a really fundamental problem with wanting a mountaintop lake of that size. Mountains, as you may or may not have noticed, are kind of pointy on the top, which means there's not a lot of room up there to put a lake. Also non-volcanic mountains tend to occur in fairly linear bands - ranges, IOW - which again limits the area on top.
I can think of maybe 4 ways to get something vaguely like the lake you want.
1) A volcano which experienced a catastrophic eruption, leaving a caldera which is then filled by a lake. Oregon's Crater Lake, formed by the eruption of Mt. Mazama about 7500 years ago, is the classic example: https://en.wikipedia.org/wiki/Crater_Lake The problem here is that there are limits on the size of a stratovolcano, which means that the resulting lake isn't going to be that much bigger than Crater Lake.
2) A valley between mountain ranges which is dammed somehow. Lake Tahoe is a good example of this: the valley formed by uplift/downdrop between the Sierra Nevada & Carson Range was dammed by volcanic eruptions: https://en.wikipedia.org/wiki/Lake_Tahoe#Geology The problem here is that it isn't on the mountain TOP, and is surrounded by a larger catchment basin that feeds it through rivers & streams.
3) Endorheic lakes, like Lakes Lahontan https://en.wikipedia.org/wiki/Lake_Lahontan & Bonnevile of the Pleistocene Great Basin. These can be at the required elevation, but aren't located on mountain tops, but within higher elevation plateaus. (The lowest points in the Great Basin are more than 1000 elevation.) Lahontan had a high elevation of ~1500 m, but is surrounded by (and enclosed as islands) mountains of 3000 m and more. They also have large catchment basins, feeding them from rivers & streams.
4) Meteor crater lake, like Manicouagan: https://en.wikipedia.org/wiki/Manicouagan_crater These aren't going to be on mountaintops, either. If the impacting body did happen to land on top of a mountain, that mountain would be removed. along with much of the underlying rock. One large enough to form a lake of the size you want would likely fill with magma to about sea level, maybe below, e.g. large lunar craters.
Bottom line is that AFAIK there's no way to get a lake meeting all your requirements.
As for your questions, #1 has been adequately answered already. For #2 you have two choices, either the water rises until it finds an outlet (e.g. Tahoe), or it's endorheic because enough water evaporates before the level rises to an outlet. #3 depends on your local climate, e.g. the Tahoe Basin and Great Basin valleys can be filled with winter fog/low clouds (as it is today), but be clear in summer.