I went over to the Apple store and they tried to help but were completely useless in the end. They aren't allowed to do the very simple option of signing in with their own account, so they had to go through all these other methods to try installing OS X.
They said the system had some Kernel Panicking, which doesn't bother me because it was working fine before this, so they scheduled an appointment for that. I tried the external drive method, I asked them if I could borrow an external drive, and they gave me one, it didn't work, it didn't power on, so they opened a brand new one right before my eyes and that one also didn't power on when plugged in. They said it was because the machine already has a disc drive in it, but it doesn't because that one was replaced with a hard drive, and the disc drive is not usable anymore.
So they asked me to call up Accounts, which then transferred me over to Technical Support, the guy stuck with me on the phone for a long time going through all these different methods but again, none of them worked. The simple answer is to sign in with an account that has OS X Mavericks on that account downloaded.
I tried downloading it from the Apple computers but login was disabled, I don't know anyone with a Mac that is accessible anytime soon so I can't sign in on theirs and download it.
tl;dr
If someone can let me use an account that was used to download Mavericks and making an account from scratch is fine as well, that would be great and I will mark that answer as the best answer.