Ooh! Ooh! I know the answer to this!! Yep - you can have six icons on one screen and 12 on another. I have my home screen pretty NOT populated and spread the wealth out to other pages. Page 2 has my more important but not used regularly apps and page three has my "spare" apps - apps that I need but not often at all (I COULD delete them but they have data that I need).
Did the ability to do random numbers of icons on one screen change with a recent update to iOS?
What about widgets? Do you still have to swipe down to access them? Or can you put them as an icon on a launcher page?