I have some unallocated space before my Windows 7 partition.
Can I move the Windows 7 partition so that the free space is after it instead than before it, and then extend the partition to use all of the currently unallocated space?
Yes you can, however, I doubt that you can using the default disk management tool.
The easiest thing you can do is to load up Gparted live cd (free) and then you should be able to move and extend the partition with no problem.
Here is a free windows partition program that also might do the job. There is another one that I used recently that worked, but I don't remember (it's on my home computer). The changes made will require windows to shutdown, then repartition, then restart.
I'm not sure if this is the software I used or not, but I assume you can find out quickly whether or not it is capable