Where is your slip located?
I think Chicago Harbors open on May 1. Some others are a little earlier, but if you are up North say in Green Bay and Door County, than many of those marinas do not get into full swing until the Middle of June.
Yes, again dependent on where you are located and how well you are protected, early Spring and late fall is when you can expect some of the more nasty weather systems. If you are in an exposed area, it may not be the best idea. I always had a personal rule when we were moored in Monroe Harbor; never launch before Mother's Day. This is a personal superstition of mine. Back in the early 80's there was a storm on Mother's day and well over 12 boats brook free from their moorings and sank. That was Monroe Harbor. We now routinely launch in mid April as well. Our Marina opens on April 1st.
Sailing in the cold yes, of course you can do it. But I much prefer cold sailing in the fall in comparison to the spring. Why? Because of the water temperature. Water temps in mid April will only be in the 40's or so. A nice 15-knot breeze blowing across 40 degree water temps and you feel it deep down in your bones. In the fall the exact opposite is true, air temps can be in the 40's and the water temps will be much warmer than the air.
I would recommend that you simply check with you harbor master before scheduling your launch.
Courtney is My Hero
If a man is to be obsessed by something, I suppose a boat is as good as anything, perhaps a bit better than most - E.B. White