Can I run a backup while I'm using my computer? If I have open files, will they be backed up?

Yes and no, it really depends. Performance will be affected during the clone (especially the first one) as CCC reads the entire source volume and writes to the destination volume. If your work is "disk bound" -- that is your applications are reading or writing to either the source or destination, then you'll notice a performance hit. If you're just reading email or writing a Pages document, then you probably won't notice the performance hit.

Affecting the accuracy of the backup task is something else that should be considered. Typically it's OK to work from the source volume while you're copying it, with the understanding that if CCC copied a file, then you open it, make changes, save it, then CCC completes the backup task, the modified version of your document is not backed up (this time around). Typically that's no big deal, the modifications will get backed up the next time the backup task runs. More importantly, though, if you're working with large files (mounted disk image, Entourage email database, VMWare/Parallels container) during the backup operation, it is possible that those large files could be modified while CCC is backing up that file. This won't affect the source file, but there's a good chance that the backup version of that file will be corrupt. For this reason it is a good idea to stop using applications that may be modifying large files.