Those 4 ports ARE the "network" ports and anyone will do.
2. If my slinklink is connected properly to the router, my network light on my slinglink should be blinking REGARDLESS if my slinbox is connected to my TV, right? I am assuming the blinking means there is data being passed to the slinglink from the router and it does not matter if the the slingbox is hooked up
Yes, the Slinglink and the router continually 'chat' to each other.
This could be a problem with your house electrical circuit (interference), since the SlingLinks will communicate each other through the electrical circuit, however, you can check the following links about the Setup.