How can I arrange the applications on each page of my app drawer?
You can press the Menu button while in the app drawer and change the view type the "Customizable grid".  Then hit Menu again and "Edit" to rearrange the icons.  It won't let you leave empty spots on a page, though.

Probably the best way to do this is simply to drag the icons onto your home screens, and then you can have a home screen for each type of app.  If you don't have enough home screens then there are probably custom launchers where you can set the number of screens.  Another alternative is folders, as you mentioned; I don't see why they wouldn't work.