Sounds like you might be in the wrong place. Firefox used to leave the downloads window open but now closes it by default when the download ends. If the download is very fast Firefox will not even open the downloads window. Try "Ctrl+J" to open the download dialog, right click on the file you are interested in and then "Open Containing Folder".

Files may be removed by you or the system and that will not be reflected in the downloads list which is basically just a log.

Tools > Options > General has some options, since mine continue to work but they go into two different places and seem to work okay. Basically those I want to keep like installs go into a permanent folder and the PDF files go into a temporary folder (firefox cache).

Firefox will automatically download files you save to a Downloads folder which, on Windows 7, would be C:\Users\<your username>\Downloads .... unless you have changed your Downloads settings in the Startup, home page, tabs, and download settings ... see the linked article for details.

See the Where to find and manage downloaded files in Firefox article for more information about what actions you can take from that interface, once the download completes.

If you actually cannot download and open or save files, see

• What to do if you can't download or save files
• http://kb.mozillazine.org/Unable_to_save_or_download_files