Okay, after more trial and error, I am finding that it is not quite the same problem as a "not closing" issue. Right after powering on my machine, and then starting FF (starts okay), it then has this problem of not being able to open any new windows of FF, even though it is the first time running it since powering on. In other words, in order for the problem of: "not being able to open another FF window being caused by not closing correctly", I would have had to close it once, which I didn't. However, It then does fix by closing it, removing it from Processes, and reopening. After doing that, it will again open another FF window. If I then close them all, restart FF, a lot of times, it still works okay, but sometimes I need to remove from Processes again. I did try reinstalling FF, but it didn't help.
UserAgent: Mozilla/5.0 (Windows; U; Windows NT 6.1; en-US; rv:220.127.116.11) Gecko/20100914 Firefox/3.6.10