I have noticed that differant makers of wood instruments have a differant time frame for putting oil in the wood bore. Varies from about one to three months. The care for the metal tuning slide varies also. Some have a case that requires the instrument to be taken apart before putting it in. To me that seems like it would cause excessive wear if it were done every day. And then the slide would be loose and need something to hold it. Seems to me that just keeping the slide clean and turning it a little each month to keep it from getting stuck would be enough?
Storing whistles with metal tuning slides in two pieces is a great idea. Moisture from your breath gets between the two halves of the slide will eventually cause corrosion which can effectively weld the tuning slide together. A loose slide is not difficult to tighten, but a frozen one is a killer to separate. I just received one of Glenn Schultz's whistles for a repair and I had to totally replace the slide.
Grease to prevent seizing is a good idea, but it can make the connection too loose. I've had success using dental floss to tighten it up again. It's made to spread under pressure.
I usually put some tuning slide grease made for Trumpets, Tubas, Baritones, and the like on it which works fine. Don't use Cork Grease it is made for Cork ad will get sticky on metal.