IPhone – Delete apps from previous Apple ID and set iPhone to factory seeting

This is most likely the carrier intentionally deactivating the old SIM.

When iPhones first came out, your current phone would normally get deactivated at the moment you purchased a new phone and told the carriers to move the service to the new phone. Now, they wait until the new phone checks in to perform the swap to account for delivery delays, backups, etc...

Your first step should be to contact the carrier and explain that you want service on both phones and arrange for which number rings to which device. They might be able to accommodate this over the phone, but if the old SIM has been deactivated in the networks, most carriers are very unwilling (or unable) to re-activate the old SIM and instead ask you to get a new SIM from them at a convenient local retail spot or by them shipping you new SIM card(s).

Apple's restore process and activation of the device doesn't affect network activation and for locked phones, you just need a SIM for the correct carrier to activate the device. iOS does not check or care that the contract is expired, not established or live for iOS activation - just that the SIM is manufactured for the correct carrier.

Plug her phone back into iTunes. In iTunes, click on her iphone, and in Options uncheck anything having to do with "Syncing Automatically". If it's not already checked, click "Manually manage music and videos".

If you want her to have a separate iTunes library, she should also have a separate login for the shared computer. This is really the easiest way to deal anything on a shared computer.

Incidentally, there shouldn't really be any reason to physically plug your iPhone into your computer anymore. You can actually set the phone (and iPads, etc) to sync wirelessly.