Hey, welcome to the wonderful world of Pocket PCs.
Your first question: In order to synchronize your iPAQ to the computer, it has to stay on. Now, if you are not synchronizing and you are just charging, then go to Settings-System Tab-Power then select the ammount of time you want your iPAQ to remain on before it automatically turns off (make sure to pt a checkmark on external power).
I am not sure about your second question but I am sure there are many cases for your iPAQ and I could almost be sure that there's a fliptop case for it.
Your third question: I think, although I am not sure that SD is a better and widely available format since ironically today was the first time I saw an SD card. They seem to have the same price. Sorry I wasn't much help on that.
Your fourth question: If you like to keep a calendar and contacts, then I recoment
Pocket Informant, which is an excellent application to replace the built in Pocket Oulook software. As for games, there are many clones of Bejeweled out there, the most famous one being
Cropped Out, which is a very good game and in my opinion, beats the old Bejeweled game.
Hope I was some, help.
Good luck.