"Is there anyway she can sync her iPhone 4S to her new iTunes account, without losing the apps previously installed in the iPhone,"
" and that way any new app she buys goes thru her account and not thru my dad's account."
She need to log into her account on the iphone. This is covered in the Settings section of the manual.
"The one thing I tried to do was to try transfer her apps to her computer using iTunes, but of course, it says the computer is not authorized, because I'm guessing the iTunes account is not the same used to buy those apps."
Apps will always be tied to the account from which they were purchased. If she is to keep the apps, then her computer will have to authorized for your Dads account. The apps will always have to be updated using your Dads account as well.