Does the Drop timer that switches you between Sora and Riku serve any in game purpose? Additionally, when does the timer start and is there a way to prolong it?
The bar in the lower right corner of the upper screen tells you how much time you have before you drop involuntarily. I believe it starts as soon as you drop, but I could be wrong. When it runs out, you drop whether you like it or not.
You can prolong it various ways. Pretty quickly, you get an item called a Drop-Me-Not, the use of which adds time to the drop meter. Also, between drops, you can purchase Drop Decelerator using drop points.
Prolonging the drop timer has the following advantages. First, it can give you more time to get more drop points and thus you can afford to buy more stuff between drops. In addition to the Drop Decelerator mentioned above, you can also get a boost to your attack, a boost to magic, sometimes an item, and other things. Second, you might want to delay dropping just because you don't want to be interrupted. I know that I prefer to keep Sora and Riku at about the same level and progress, so if one of them gets too far ahead I might want to use some drop-me-nots and manual dropping to play the other one more.