Hello. When you are downloading Firefox, are you saving and installing or just running the program?

Follow the steps and if asked, save the contents to Desktop. https://www.mozilla.org/en-US/firefox/new/

Try this:

• download the Firefox installer and save the file to the desktop
  *https://www.mozilla.org/en-US/firefox/all/
• start the installation with a double-click on the desktop icon

It is possible that security software (anti-virus, firewall) is causing the problem. Try to disable security software temporarily to see whether that makes a difference.