Sounds like a pinch alright.

By Refreshing Firefox, you created a new blank profile that was then populated with your bookmarks and other personal data. Only Firefox uses these profile files, so this does not affect your system as a whole.

If you've used System Restore, that will have rolled more than Firefox back, as it literally tries to restore your whole system to a previous state, warts and all. Depending on your settings, this will involve system files but also application and user-created files - this can mess with software from antivirus to browsers to drivers, games and more. Since you were having trouble before you took these steps, there is possibly an underlying cause here.

System-wide connection trouble and the inability to install new browser versions from the internet can be a symptom of malware infection (but also corrupt browser caches, bad system connection settings or a haywire firewall).

To make sure you're safe and to restore your system to a functioning state, I would suggest the following:

• Make backups of all personal data. Do not overwrite previous backups in case the new backups are corrupted or infected with malware.
• Run Norton LiveUpdate manually to make sure your antivirus software, firewall and virus detection databases are up to date. Once your program version and files are current, run a system scan (or whatever Norton calls the most comprehensive scan).
• Complement the Norton scan with other scanners: https://support.mozilla.org/en-US/kb/troubleshoot-firefox-issues-caused-malware#w_how-do-i-get-rid-of-malware_2. Advanced malware will be programmed to limit your access to help and repairs (including scanners, hence multiple ones are a good idea when troubleshooting), plus it may be using your system's resources for itself.
• Reinstall the drivers for your hardware from manufacturer websites (not Windows Update, not third-party websites). They may have gotten updated previously and then rolled back due to System Restore, which may have left new and old files and incompatible settings mixed together.
• Repair (program or installer functions permitting) or reinstall any software having issues. System Restore may have messed with them the same way it messes with drivers, mixing old and new and leaving the software unable to function and determine its correct state.
• Check system connection settings (proxy etc.). You'll have to use Microsoft's instructions for this one, as I don't have experience working with Windows 10 system settings. Basic: http://windows.microsoft.com/en-us/windows-10/getstarted-why-cant-i-get-connected. Advanced: http://windows.microsoft.com/en-gb/windows-10/fix-network-connection-issues.

Protection, drivers, software, settings should be the marching order when rescuing a system. Preferrably use another, working system to retrieve the necessary installers and instructions.

Boot the computer in Windows Safe Mode with network support (press F8 on the boot screen or hold down the Shift key) to see if that has effect.

• http://www.bleepingcomputer.com/tutorials/how-to-start-windows-in-safe-mode/
• http://www.7tutorials.com/4-ways-boot-safe-mode-windows-10

Do a malware check with several malware scanning programs on the Windows computer.

• https://support.mozilla.org/kb/troubleshoot-firefox-issues-caused-malware

Please scan with all programs because each program detects different malware. All these programs have free versions.

Make sure that you update each program to get the latest version of their databases before doing a scan.

• Malwarebytes' Anti-Malware:
  http://www.malwarebytes.org/mbam.php
• AdwCleaner:
  http://www.bleepingcomputer.com/download/adwcleaner/
  http://www.softpedia.com/get/Antivirus/Removal-Tools/AdwCleaner.shtml
• SuperAntispyware:
  http://www.superantispyware.com/
• Microsoft Safety Scanner:
  http://www.microsoft.com/security/scanner/en-us/default.aspx
• Windows Defender:
  http://windows.microsoft.com/en-us/windows/using-defender
• Spybot Search & Destroy:
  http://www.safer-networking.org/en/index.html
• Kasperky Free Security Scan:
  http://www.kaspersky.com/security-scan

You can also do a check for a rootkit infection with TDSSKiller.

• Anti-rootkit utility TDSSKiller:
  http://support.kaspersky.com/5350?el=88446

See also:

• "Spyware on Windows": http://kb.mozillazine.org/Popups_not_blocked