I don't know if it was mentioned, but as airspeed increases, rudder effectiveness increases. If you're not slightly reducing rudder pressure after rotation, you're going to get coordinated and aircraft control gets more dfficult. I don't know how the airlines do it, but we get the autopilot on as soon as we can to reduce workload.
As far as the autopilot goes. Always take a second to see what mode you're in before pressing anything. The FMA is what you're looking at to see the mode. Never just punch a button and assume it did what you pushed. I usually verbally say what mode I push aloud. "NAV", "FLC", "VNAV" and etc... Then verify that it's doing what you want it to by saying "NAV captured", "Speed captured", or "Path captured".