Many of these questions and answers appear in The Snow Booklet by Doesken and Judson, U.S. Forest Service Alpine Snow and Avalanche Project, retired. No, it can snow even at incredibly cold temperatures as long as there is some source of moisture and some way to lift or cool the air.
Snowflakes are agglomerates of many snow crystals.
No routine measure of snowflake dimensions are taken, so the exact answer is not known. Snow forecasts are better than they used to be and they continue to improve, but snow forecasting remains one of the more difficult challenges for meteorologists. One reason is that for many of the more intense snows, the heaviest snow amounts fall in surprisingly narrow bands that are on a smaller scale than observing networks and forecast zones. This is part of the fun and frustration that makes snow forecasting so interesting. I recall people telling me that there is 1 inch of water in every 10 inches of snow that falls.