Firefox 3.6.* is compatible with Win XP 64.

You may have an add-on that is incompatible with Firefox 3.6 and is causing the problem. The first thing you need to do is reset your useragent string, it has been modified by the Megaupload add-on and is causing your version of Firefox to identify itself as Firefox 2.0.0.12. To do that see How to reset the default user agent on Firefox.

When you next try to upgrade Firefox, it may now automatically identify any incompatible add-ons for you and disable them since it will realise that you are using Firefox 3.6.8 and not believe you have 2.0.0.12. If it does not, try the procedure in the Troubleshoot extensions, themes and hardware acceleration issues to solve common Firefox problems article.

You have a corrupted user agent (Firefox 3.5.5):

Mozilla/5.0 (Windows; U; Windows NT 5.2; sk; rv:1.9.0.5) Gecko/2008120122 Firefox/2.0.0.12;MEGAUPLOAD 1.0

You have or had an extension installed (MegaUpload toolbar) that has changed the user agent from Firefox/3.6.x to Firefox/3.0.x.

You can see the Firefox version at the top and the user agent at bottom of the "Help > About" window (Mac: Firefox > About Mozilla Firefox).

You can check the general.useragent prefs on the about:config page.
You can open the about:config page via the location bar, just like you open a website.
Filter: general.useragent
If general.useragent prefs are bold (user set) then you can right-click that pref and choose Reset.

See Web sites or add-ons incorrectly report incompatible browser and Find what version of Firefox you are using

See also http://kb.mozillazine.org/Resetting_your_useragent_string_to_its_compiled-in_default

I change it (useragent) but I have still same problem. Now it is not bold, my version is 3.6.8 version of browser and version of superagent settings but the problem remain the same

after 1 minute of browsing I can't... log in router: too many open sessions

I trie to switch off all add-ons, but situation remain the same. My opinion is that firefox 3.6.x not support win xp 64-bit. it is stupid...

If one version works and the next version doesn't work then a possible cause is security software that blocks the new version.

Remove all rules for Firefox from the permissions list in the firewall and let your firewall ask again for permission to get full unrestricted access to internet for Firefox.

See Server Not Found - Troubleshoot connection problems and Configure firewalls so that Firefox can access the Internet and http://kb.mozillazine.org/Firewalls

Create a new profile as a test to check if your current profile is causing the problems
See Basic Troubleshooting: Make a new profile

There may be extensions and plugins installed by default in a new profile, so check that in "Tools > Add-ons > Extensions & Plugins"

If that new profile works then you can transfer some files from the old profile to that new profile (be careful not to copy corrupted files)
See http://kb.mozillazine.org/Transferring_data_to_a_new_profile_-_Firefox