I see that you have some questions before installing your new Slingbox, let me help you with that.
The Slingbox needs to be connected to an AV source and to a router in order for it to stream audio and video over the Internet.
Once we get her a new U-Verse set-top box, do we have to connect it through an extra TV?
It is not necessary to connect a TV to the new cable box in order for you to view the Slingbox content, however it might be needed while troubleshooting the AV source.
Do I have access only to the programs recorded through the STB connected to the Slingbox, or to anything on her DVR? (We often watch the same shows and I don't want to hog space on the DVR by recording something she's also recorded using another STB if I don't have to.)
If your Slingbox will be connected to the DVR, you will be able to watch and record programs using the DVR options on the remote. However, if your Slingbox is only connected to a separate set top box, you will only receive the content available from that device.
You can use the following link to retrieve the instructions on how to connect the Slingbox to your AV source:
I hope that helps.