Can I use both back wire inserts and 1 of the side screws? In other words can I use 3 hot connections and 3 neutral connections on 1 outlet?

You can do that if they aren't all actually hot wires. What you are probably trying to do is have one hot wire on the bottom half and a hot wire on the top half and the other wire is actually a traveler that goes to the next receptacle. If you have any questions about this, please reach out to us at (800) 223-4162 8:00 a.m.-8:00 p.m. EST or E-mail:customer.support@legrand.us and we would be happy to help!