My rough turned bowls are in my unheated/uninsulated garage. They sit on shelves and are definitely in below freezing temperatures. Have had 2 liter bottles of soda explode in my garage in the past. Having a climate controlled environment has to be better for them, but I can't say that temperatures have caused issues or defects.
I've brought in frozen logs from outside in winter and turned them, and can say definitively that frozen shavings hurt a bit.
And don't get rid of the flat woodworking stuff - you might someday want to do some segmenting or staved vessels. Plus, those flat surfaces make for good storage!