AT&T U-Verse TV is separate from the internet for all practical purposes, what you do on one will not affect the other. What goes on on your home network shouldn’t be adversely affected by what goes on on the internet. You don’t even need the internet for a home network to exist.
The WDTV Live, Live Plus, Hub, SMP or Play do not have the ability to record. With U-Verse the only way I know to save/record a show is by using their equipment and it doesn’t allow you to move it from their equipment. I still have a DVR myself and have never plugged it in since I got U-Verse. You could probably record from their set top box if you really wanted to, but I really haven’t felt a need for that.