As far as I know as long as the drive isn't encrypted, you should be able to right click on the top level folder (in this case you would want the user folder for the original user) and change to read and write so that you can access the files within. as far an I know the files you want are in "AppData\Local\Mozilla\Firefox\Profiles" and "USERNAME/AppData\Local\Google\Chrome\User Data"
As for installing windows or not, if it is identical (down to the specs and model) you could try to boot from that drive using del or f2 to get to the bios and select that drive as the boot drive.
If the issue is that the drive doesn't boot properly, then yes you will have to install windows on the other drive (it sounds like you had 2 installed)