What I'm getting from your post, is that you're having problems with the Slingbox streaming.
Based on the information that you're providing, we can tell that this could be a Double NAT problem affecting the Slingbox connectivity.
What I would recommend, is to connect the Slingbox directly to the Verizon router and set it up there. After that, you can connect the Slingbox back to the Time Capsule (acting as a bridge).
This might extend the wireless signal, without affecting the Slingbox.
Besides that, and to discard the Slingbox as a potential issue, you can connect, set and test the Slingbox, connected to the Verizon router only.
You can also check the following link related to the issue that you're having: