If I sign a transfer who is a true sophomore, do I need to burn one of my 2 redshirts, in order for him not to lose a year of eligibility, while he sits, or will he automatically redshirt, and I can use the redshirts on 2 other players, if I'd like?

He'll automatically be ineligible for the upcoming season. It won't affect your redshirts. You just have to have an open scholarship on your CURRENT roster to sign him. He won't lose a year of eligibility.