Do I need "Install OS X Mavericks" after I've upgraded? I have "Install OS X Mavericks" in my Applications folder even though I installed it a while ago. Does the system require this for utilities or something, or is there any reason to keep it around?
No. If you have no use for it, delete it. The OS doesn't need it any more and if you want to reinstall OS X you can use the Recovery HD, so there's generally no need to keep it.

The post-install script will even delete it for you if its found in the default location as far as I know, but if you still have it there's no problem deleting it yourself.