2) "Controls: full/free movement in the correct sense" check???"
This check had already been completed before I got in the a/c, although I agree, for safety purposes it should have been checked again before t/o. I assume that as the piper control surface are controled via "mechanical" means, once tested they should not require double checking surely?
I was taught to do a full + free as part of the initial checks, *and* just before the pre take-off vitals. Part of the point of the full and free check is to check that the controls haven't been obstructed by luggage or passengers. It's not much use checking it before you've got yourself and all your kit on board!
5) "I like to squeeze on a little brake just as the aircraft starts to roll as you start the taxi to make sure the brakes are in fact working"
This is another good point, that I was once told by an old instructor that I once flew with. I remember him actually failing a student who was on a solo circuit test because he didn't do this. I did it a few times with other instructors but was told something along the lines of "assuming the brakes don't work once you've started taxing, just idle the mixture to shut off the engine...problem solved".
I think it makes more sense to check the brakes as soon as you start rolling - what if you get up to taxi speed before discovering a problem. It's too late to do anything about it then...
6) "When doing the runup don't rush. When you checked the carb heat it was only on for 2 or 3 seconds"
I was under the imrepession (as you've stated) that if there is no ice the RPM will drop. If I watch the RPM drop after 2-3 seconds and running smoothly, what else do I have to wait for?
If you apply carb heat and there is carb ice you'll still get an RPM drop. What will then happen a few seconds later is that the RPM will increase as the ice melts. If you only check it for a couple of seconds, you've verified that the carb heat control works, but not actually checked for carb ice.