There is a shuttle that takes ~10 minutes between T1 and T3. With an 1:16 connection, you have plenty of time.

However, as you seem to not be Elite, you'll have an E-seat (behind Row 21) in the first flight and most likely Boarding Group 3/4 on the 2nd flight.

In other words, space for the overhead is going to be tight. Best to check it all.

Pre-check is irrelevant with respect to your outbound situation.

For your inbound, unless you meant Global Entry, no difference.