Did Firefox restart in safe mode. You said

 I have tryed to run firefox in safemode without results,

Try by using the option from the Fifrefox button to restart with addons disabled do you see a warning dialogue and then it restarts after you accept that ?
Have you tried starting Firefox in safemode. Close Firefox then hild the return key as you click on the Firefox icon.

• see Diagnose Firefox issues using Troubleshoot Mode

What happens when you select text on a webpage e.g. this forum and right click the selected text. You should see an option to search the web.

Note the searches will use the search engine selected for the searchbar, normally that is Google.

• Search bar - add, change and manage search engines on Firefox

It is normal for the address bar to show search or enter address until you have entered something.

I may understand the location bar misbehaving but I do not see why the cursor should move there by itself !

Can you confirm you have actually restarted Firefox in safemode ?

Try to open a new window to see if that helps/

Create a new profile as a test to check if your current profile is causing the problem.

See "Creating a profile":

• https://support.mozilla.org/kb/profile-manager-create-and-remove-firefox-profiles
• http://kb.mozillazine.org/Standard_diagnostic_-_Firefox#Profile_issues

If the new profile works then you can transfer files from a previously used profile to the new profile, but be cautious not to copy corrupted files to avoid carrying over the problem

• http://kb.mozillazine.org/Transferring_data_to_a_new_profile_-_Firefox

I have tryed to restart firefox in safemode with addons disabled, and the problem isn't solved. When i restart it i see the warning dialogue box, and i accept that.

What happens when you select text on a webpage e.g. this forum and right click the selected text. You should see an option to search the web. Note the searches will use the search engine selected for the searchbar, normally that is Google. - I'm not sure what this means exactly? English and computer "language" isn't my strong side :)

It is normal for the address bar to show search or enter address until you have entered something. - I know it's normal, but this is what happens when i for example choose www.google.dk or www.facebook.com. After i type the webadress in the address bar, it enters the website and after that, the adress bar doesn't say www.google.dk og www.facebook.com it says search or enter address and the curser now stays in the address bar, so when i want to scroll down the site on facebook by using the "down"arrow it instead shows addresses that i have visited in the address bar, as when i click the address bar and use down arrow to se which sites i have recently visited. Then i have to click on the facebook page somewhere, and the curser then dissappears in the address bar for a little time, so i can use the arrowkeys again to scroll down, but after a short time the curser jumps back to the address bar again.

Current Firefox versions also use the search engine that is selected in the search bar (click the search engine icon to open the drop down list) on the Navigation Toolbar for searching via the location bar and for the search engine on the about:home page.

If you want to use different search engines then you can install the Keyword Search extension to specify with search engine to use for the location bar and which search engine to use for the about:home page via the Options/Preferences windows of this extension, accessible via the about:addons page (Firefox > Add-ons).

• Keyword Search: https://addons.mozilla.org/firefox/addon/keyword-search/

Firefox wouldn't move the focus from the web page to the Navigation Toolbar without pressing a key like F6 or Ctrl+L.

Did you try to close and restart Firefox or reboot the computer?

Start Firefox in Safe Mode to check if one of the extensions (Firefox/Tools > Add-ons > Extensions) or if hardware acceleration is causing the problem (switch to the DEFAULT theme: Firefox/Tools > Add-ons > Appearance).

• Do NOT click the Reset button on the Safe Mode start window.

• https://support.mozilla.org/kb/Safe+Mode
• https://support.mozilla.org/kb/Troubleshooting+extensions+and+themes