Depends on the type of automatic carwash. Definitely avoid the ones with the hard plastic brushes, those are murder on the clearcoat. The ones with the soft cloth strips probably aren't much better. They might be OK when new, but most get so much dirt on them they'll scratch too. There's a few auto washes near me that are the "touchless" type. It's basically an automatic high-pressure wash. They do a pretty decent job, IMO. I use them all the time in winter.
Now and then the locks will freeze up after a car wash, so it pays to keep some de-icer on hand. Note: keeping it in the glovebox isn't as handy as it sounds at first. Chicken and egg problem if you're locked out of the car.
I have an unheated garage and it stays pretty cold here in the winter so melting, condensation and rusting aren't too much of an issue. Also we don't get much salt on the roads (too cold to have much effect generally), but we get a lot of sand/gravel. My main concern with keeping the car clean in winter is to get that grit off so that it doesn't damage the paint or come off on our clothes if someone brushes up against it.