skype disappears a day after download

Did you actually install the program or simply download it? If you just downloaded the program then you need to go to your temp file and run the install process. When the program installs onto your program it will save itself to your Program Files folder on your C drive.

Agreed. I've downloaded and installed Skype a few times - as have friends and there's been no problem with Skype procedure.

If you don't actually install the program (following the procedure), it can't be run. Also make you sure your virus protection software isn't blocking it somehow.