Please forgive my ignorance or naiveness with this question. But, should I be concerned when trying to run 32 bit applications on a 64 bit operating system?
Yes, they will surely run smoothly without any problems. 64 bit OS are more advanced than 32 bit OS and they are more capable of running applications lower than 64 bit.
There would be no problem in running a 32 bit application because the problem will only occur when you are running a high bit application against a low bit memory system. The same thing goes for other kind of software as well.