Can I save all the contacts in a Phonevite to a new Phone Group?

Yes. After all the contacts have been selected and loaded, and prior to sending the Phonevite, you can click on Save all Contacts to a New Group under the 2. Add Contacts section.

Alternatively, after you have sent the Phonevite, you can go to the My Phonevites page, click on the Call Details for the Phonevite in concern, and from the Call Details Page, from the More drop-down menu, click on Add to Phone Book.