I would like to auto login returning members who have registered on the site before using Solspace User. Is this possible? Is there an auto login tag/plugin I could use when they register?
User employs the EE member login functions. There is nothing present that overrides them. You will also use the stock https://ellislab.com/expressionengine/user-guide/add-ons/member/#login-form-tag The most you can do is expose the "Remember Me" checkbox.
I use this all the time and it works like a charm (Solspace User and autologin). Remember "In order for this feature to work you must be set to use “cookies only” and not sessions (source)."