I've been in Vegas over the time period before, and in fact will be again this year (Dec 26-28).
Most hotels in Vegas have outdoor pools, and most of them are shut down over the "winter" months. Somehow, coming from the Canadian prairies, "winter" in Vegas just isn't the same as "winter" here. I did find that a light jacket was needed in the evenings, as it does cool down a LOT. The first time I was there in December it was horribly windy the first day or so, and as a consequence was quite cool because of that.
As for the hotels you've mentioned, all 3 are fine hotels, which one is right for you depends on what you'd like. Are you a buffet person? The Venetian doesn't have one, the Bellagio's is one of the best in Vegas and the Mirage... well it has one. Last time I was there I'd rate it about a 6/10. I wouldn't rate their coffee shop much higher, but I understand their other restaurants are quite nice. Both the Bellagio and the Venetian are on the higher end of the luxury scale, and as a consequence have a number of very nice (and higher priced) restaurants. And shops.