No internet access after booting or reset (Windows 7)

Restore to a date prior to when you had the issue. Just because you uninstalled / used malware cleaners and an antivirus doesn't mean that whatever got installed didn't alter system files, registry stuff etc.

If this is not an option for you explain why, and we can give some other tips / help.

Do the restore in safe mode.

Then your safest best is a full reformat imo. Virus's, rootkits, malware etc can cause nasty dmg that is irreparable sometimes. Back up all of your important data. Make sure you have any and all license keys that you need (Microsoft office, Windows 7, etc etc.) And do a complete re install. The only reason I suggest this is your issues came up after installing some malicious junk.

Other things that should be asked too.. Are all other devices working fine on the connection? Make sure to rule out that it's not a coincidence of the install and something else going on with your internet, router, modem.

I personally wouldn't install windows 10 yet.. Ive tested it on my work laptop and ran into a lot of issues with it. As with most OS's the first year or so is going to be filled with bugs until they release future patches.

Did you fully uninstall Avira and any other antivirus software before attempting to install AVG?
Asked this above but want to ask again. Are all other devices working fine with the internet? Wired and wireless?
You shouldn't lose internet connection installing or uninstalling avg. That is not normal behavior.

Things to try:
1) create rescue disk (avgs rescue disk is fine) from another unaffected machine. Boot your problem machine from the rescue disk and run the scan. I like this because you are finding the virus/malicious crud before drivers, the OS etc load. Sometimes these bugs are rooted deep and don't get cleaned up properly.
2) You can also run your malware cleaner (I recommend malwarebytes) and Antivirus scans from safe mode. You should always run them here when having issues like you are. Again, it's only system critical things that will load.
3) If it's a pre built machine and you still have the disks you can try to run a system repair from the original disks. It may or may not correct files that were damaged.
4) Reinstalling your net card drivers. (will be under your mobo's website, or if it's a pre built machine hp, dell etc you can find them on their website with the make/model of your unit)
5) Could be that the port your plugging into is failing. A quick test would be plugging into a different port if available, or a stand alone net card.

Everything from your posts leads me to believe that it was some sort of malicious software that is causing your issues thou... To much a coincidence that you started having the issues after those things were installed/uninstalled. Keep in mind... Just because you uninstalled and ran scans that does not mean that the system files etc were not altered, damaged or changed by w/e caused it. Hence why I recommended a fresh install. Root kits, Some viruses, Trojans etc can be very nasty and cause irreparable damage.