Figured I'd post my results in case it helps anyone out in the future.
Last night I did a clean install and did a full Time Machine restore during installation, and as before, that failed and I could not launch the Simulator.
After that, I tried a clean install then only restoring settings/documents from my Time Machine backup, only unchecking Applications, and this seemed to work just fine. I downloaded Xcode 7 and could open the Simulator no problems. I also have all of my settings/files in the new installation so all I have to do is install some software which shouldn't be too difficult.
So something that was tied to the previous installation of Xcode must have been causing this odd conflict. I'm just now glad I can finally take a look at why one of my iOS apps is crashing on iOS 9 since I can now debug iOS 9.