What .exe files are you downloading?
Are you sure they are from a safe source and that Norton isn't finding some malware in them and therefore removing them?
If you have deleted Norton, do you have another AV running which may be doing the same thing? (like, is Windows Defender still activated and there is spyware in them?)
Unless you are certain these files are safe, its unwise to deactivate your AV package, and certainly not for long. Run your AV package, update it, and run a full scan.
Where are you trying to save the files? is it in your user area?
Is Firefox your browser? I don't use it but found this long read - http://www.sevenforums.com/browsers-mail/135139-auto-delete-downloaded-file-3.html which sounds similar.