App store thinks I have a bunch of apps installed after I reinstalled my mac
The App Store app only consults spotlight for your Mac to determine if an app is installed, so we can rule out any account problem (and you can sign out of the account from the store and reboot to confirm this on your system.)

So, since you have a few apps, you can focus on one app app and check spotlight to see if there are copies of it elsewhere than in /Applications.

Some helpful hints:


be sure to eject all disks except for the one drive you are booted from
consider rebuilding your spotlight index after checking for apps on other volumes by ejecting the volumes