After Malware attack and PC reset, "Can't Connect to this Network"

I recently downloaded malware while searching for a CD-key generator for Neverwinter Nights 2. After the initial download it began downloading unusual programs, deleting programs I had installed, like Chrome, interfering in my ability to download Malwarebytes (did not have a virus protection installed beforehand), and taking over network controls through a group policy. I quickly re-installed my OS, Windows 10, twice and cannot provide specific file names of the malware. First re-install, it kept old files. After getting the pictures, txt (notepad and a spreadsheet), and emulator files on a external, I deleted everything in window.old (it said there were system files in there, but OS installation said that folder would go unused so I deleted everything). Second re-install, i made sure I chose the clean install option. Now I am encountering "Cannot connect to this Network" error when I try to connect to my home's Wi-Fi. I have a roommate who's Apple Mac Chrome Application started saying your connection is not private on familiar web pages and my own work laptop became unable to maintain a stable internet connection to the home network it had no trouble with the day before. I have tested my laptop's network adapter at a friend's house and encountered no such issue. This connection issue only started after re-installing Windows 10 after the malware infection  (I can't say if my WiFi connection was fine during the malware attack as it was slowly degrading during my attempts to remove malware by an anti-malware program and the group policy control prevented my from interacting with network controls). I have not touched router as we do not own it; it is U-verse router from AT&T. I had hopes that a technician from there might be able to help, but after reading other virus topics here, no longer hold that hope. 
Is this a stubborn malware issue that persisted somewhere a factory reset would not erase?

Is our router infected by this?

Please feel free to request more information. I'm sorry most of my information is qualitative and not very specific or quantitative. 

Operating system: Windows 10

While not condoning one bit your downloading of CD-key generators, I will ask if you have already tried the complete removal via Control Panel --> Device Manager of your network adapter[s]; Restarting, and allowing Windows enough time to reinstall necessary drivers from your hard-drive first?

Edited by RolandJS, 10 November 2016 - 09:11 AM.

While not condoning one bit your downloading of CD-key generators, I will ask if you have already tried the complete removal via Control Panel --> Device Manager of your network adapter[s]; Restarting, and allowing Windows enough time to reinstall necessary drivers from your hard-drive first?

It's a legitimate copy. I just don't have the bloody cd key with me in this state. 
Anyway, I was afraid of completely uninstalling my network adapter but I did it and that seems to have fixed it. Looks like whatever I downloaded is gone for good. 
Thank you for the quick and simple assistance. I felt like I was going out of my mind.