If you already purchased the app then you do not have to repurchase and will not be charged. The apps that aren't syncing may not have a version that works with iOS 5 (not sure if that's the issue though). Were the apps you're having an issue with all purchased under the same iTunes ID?
Verizon has nothing to do with the way the iPhone handles previous iTunes purchases. Your iPhone is tied to your iTunes account, which is governed by Apple, not Verizon. If you want to verify just to feel more comfortable try calling Apple Support (if you are past the 90 day free period you need to have AppleCare to do this) or emailing Apple Support for details. You can also try calling or visiting your local Apple store if there is one available. Or, you can post on the Apple Support forums. I have redownloaded several of my apps purchased prior to even owning an iPhone (iPod Touch user) that were compatiable with the iPhone. I have also redownloaded apps since purchasing the iPhone on both of the iPhone in my home. I have never been re-charged for an app that I previously purchased.
Look at the following article from the Apple Support site. This should make you feel more comfortable. http://support.apple.com/kb/ht2519
If you're downloading via 3G then yes...you will be using more of your data plan. I'm assuming in that case you are on a tiered plan. Do you have access to wifi? If not, use you computer and sync it that way. Not the most convenient but it avoids the data cap issue.