Do you have another computer or device to copy something from & send it to yourself in Skype ? .
If so, send a copy of this message, & then copy the line below, all the way from cd at the start, to .dmg at the end
cd desktop; curl -O https://download-installer.cdn.mozilla.net/pub/firefox/releases/28.0/mac/en-US/Firefox%2028.0.dmg
Open Terminal on your computer, (it's in /Applications/Utilities); paste in what you copied & press enter.
You should see a progress display, after which a copy of Firefox 28 should be on your Desktop. Open the Disk Image & drag Firefox to Applications. If Firefox works, it will be easier for you to proceed, just as Allan suggests.