I accidentally got a mate. Could I get rid of my mate in the game?

Once you get a mate, you can't get rid of them. But, you can keep them in the den via shield button. Use howl and look for the home icon, run to it, enter, and use Shield icon. It'll make the mate and babies stay in the den.

I don't think you can. Your mate sticks by you. But if you don't want them hunting with you, you can restrict them to the den by using the shield button.