MacBook Pro 13" Retina early 2013 lost power during update

Hello, I have been trying to fix a 13" MacBook Pro Retina early 2013 that was shut down during an update by a friend. I used Disk Utility in recovery to erase the hard drive and then reinstalled OS Mountain Lion, but for some reason it seems to have lots of problems. Most websites can't be reached, I got YouTube to open, but the page was blank, minus some grey boxes where video thumbnails should be, Apple.com's retail area seems to work fine but, I can't even get into the Apple Support pages. I can search Google successfully but, can't open the page of any results. I ran Disk Utility and checked permissions, got like 10-20 errors, ran repair permissions and it said it fixed them, but when I checked again, it looks like they're all still there. I tried to install all software updates available, all but iTunes worked, for that I get a download error 102. I also noticed a separate HD or partition that's only a few GB, I'm assuming that's where the recovery stuff is located, I feel like that may be corrupted and, that's why I have issues after what seems like a successful install of Mountain Lion. It looks like most other stuff works, but Safari is the thing I use the most, so it's basically useless without it. Is there a way I can get Mojave, or whatever the most recent OS is that will work on this particular MacBook, and do a clean install with that?