Lots of good advice here.
According to this carb ice chart, the likelihood is pretty faint at 32C, but not totally impossible. Don't forget that the temp in the air was likely cooler.
http://www.faa.gov/ats/afss/newyork/ENROUTE.htm
You may have been taught to take off the carburetor heat on short final so that you have one less thing to forget if you have to do a go-around. I have several memories of being whacked upside the head whenever I did not push it in before advancing the throttle.
As for your throttle movement, unless you have taken to advancing it more rapidly solo than with your instructor, the accelerator pump should not be an issue.