Let me help you with those questions you have before purchasing your Slingbox.
I don't have a sattelite box/dish, DirecTv, or anything like that. Just a cable line that comes up from the floor and screws directly into the tv to pick up.
Since you do not have an Audio/Video source because you get your TV content with a coaxial cable, you will be able to use the Coaxial input on the Slingbox PRO-HD and scan the channels available. The Slingbox SOLO does not have a coaxial input, so it will not be the right Slingbox for you.
Also, my router is in a different room than the TV i'm hoping to hook up to the slingbox although the router is fairly powerful and kicks signal back to my room pretty well.
Also, since your router is not near your TV you will need to use SlingLinks or a wireless bridge for the installation.
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I hope that helps!