If you get a message saying that the log file should be there but it's not, and then you run the application but the files start disappearing, it seems a zealous anti-virus is removing any file you try to access.
Also, when you say "the icon is missing" do you mean that the icon has changed to some generic Windows application, or that the .exe file has disappeared?
I would suggest you try installing it in a different directory, and maybe leaving a window with the files on sight to check at which point the content changes.
If nothing works, the next thing to do would be to try in a different computer, just to confirm that the issue is not related to the installation process.
Let us know how it goes.