best time to transplant this tree.....pro and con of doing it in summer .....living in northern ohio

I would think your best chance of successfully transplanting a hemlock would be in winter when it's dormant. Late fall after a few frosts, at the soonest.

If it's very small, say less than a foot tall, and you can put it in a pot, keep it in the shade over the summer and water it diligently, you might get lucky with it. They grow a long, deep root as I recall so be sure you get the whole thing.

Welcome to Garden.org, @buster !

How large is the tree? Summer would probably be the worst time to transplant it, because of the added stress of the heat. I would think that spring would be the best time to transplant it, but the larger it is the more difficult it will be to successfully move it. As Elaine said, they do have a deep tap root; you could try "root pruning" the tree in the fall and then actually moving it in the spring.