I have Tor Browser installed on my Mac. If I erase it will there be any trace of it on my hard drive? Or do I need to back up files and format? I have flash memory.
Also is it a good idea to run a VPN through your router since Tails doesn't let you run a VPN?
You do not need a VPN. When using tails you're secure enough and VPN can actually cause you to be less secure due to data leaks,etc.
Tails has good documentation on why not to use VPN. Read it. :)