Would this work well on a cement?

Yes, I believe it is stable enough to work on cement. It might skootch a little, though. We have it set up on packed down mulch and it doesn’t move a bit. I think it would be fine on cement.

Can you leave this outside except during the winter? It will be a pain to keep setting up.

Set up / take down is pretty quick and simple. However, if you use multiple times daily and wish to keep up, it takes a long time for net to degrade. I'm in CA (very little rain) and have had mine in back yard up for before COVID 19 start (so over 18 months). The frame is still good, but the net is showing some tears where they fit into pole. I suspect this is more due to the number of balls "stretching" it back. If you take it down after each use, this wear most likely distributed more evenly. I'm back to buy another, for when the net finally does tear off. For another user asking, I use four concrete blocks (I think called "retaining blocks") on the legs to keep net from tipping over or moving.

Is this suitable for/capable of use by adult hitters?

Yes, definitely suitable for adult hitters.

I need this tee for 12U softball will it work?

This doesn't come with a tee. If you need the net, yes it will work great for your 12U softball needs. I'm still using it with my 14U all star baseball team. It's been awesome!!