Would I be able use the Transfer my Data application to send data from my Windows Phone or keep my data in sync across multiple devices?

No, this app is typically meant to be used when the user needs to transfer all the bulk data from his/her old phone to the new Nokia Lumia with Windows Phone 8.

It is designed for extracting data from an old device and injecting it into Windows Phone 8 data repositories. It does NOT send data from the current device to another.

Repeated transfers result in data duplication (visible only when editing or deleting contacts).