It sounds like you probably have covered most of the suggestions in this article: Firefox can't load websites but other browsers can.

When you find time to reinstall, here's how I suggest doing it. This actually only takes a few minutes because you are re-using your existing settings.

Clean Reinstall

We use this name, but it's not about removing your settings, it's about making sure the program files are clean (no inconsistent or alien code files). As described below, this process does not disturb your existing settings. Do NOT uninstall Firefox, that's not needed.

(A) Download a fresh installer for Firefox 45.0.1 from https://www.mozilla.org/firefox/all/ to a convenient location. (Scroll down to your preferred language.) For maximum plugin compatibility, choose the 32-bit version.

(B) Exit out of Firefox (if applicable).

(C) Rename the program folder as follows:

(32-bit Firefox on 64-bit Windows)

C:\Program Files (x86)\Mozilla Firefox

to

C:\Program Files (x86)\OldFirefox

(Other systems)

C:\Program Files\Mozilla Firefox

to

C:\Program Files\OldFirefox

(D) Run the installer you downloaded in step (A). It should automatically connect to your existing settings.

Any improvement in loading pages?

Note: Some plugins may exist only in that OldFirefox folder. If something essential is missing, look in these folders, but to avoid introducing problems, please don't just copy everything over:

• \OldFirefox\Plugins
• \OldFirefox\browser\plugins

Hmm, your screen shot is for http://www.google.com/ which should redirect to https://www.google.com/ -- can you go directly to the secure site using that second link? (Probably not, but I thought it was worth asking in case the problem involves redirects being stripped or blocked.)

Unfortunately, none of these advice didn't help. I did everything from the first post and tried to access https version - still nothing happens.

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

Since I am a helper on GeeksToGo site, I also checked FRST, OTL and Gmer logs from this computer along with scans. Effect - no viruses found, but it seems that I found a solution because now everything is okay. It looked like there was a problem with connecting using HTTPS, so I did a thing from this topic:

https://support.mozilla.org/en-US/questions/959914 

Changing autofill for false did the work. After a while I changed it back to true and it still works for a few hours right now.