The cable company does own the box they mounted to your house as they paid for it and installed it.... HOWEVER, they do NOT own the lines that run from inside that box, to your cable jacks inside your house. The only cable the own is the one running from the street (or pole) to inside of that box...
Yes, the Satellite guy can in fact use those lines... some guys open the box up and remove a portion of the lines.... however most of the boxes have a designated area (usually actually marked in there) that says something like "Customer Access" ... so even if they dont want to open up the cable companies portion, they can just snip the line before it runs into that panel and put their own fittings on them and do their connections on YOUR side of the box.
Be prepared though, any which way, you WILL have some cables on your house... but only from the Dish to the cable box outside...
We actually had a whole big dispute with the local cable company in my area after one of their techs came to install cable internet on a home that we just did a New Connect on like 3 days prior(this had happened on several different occasions prior to this, by the same cable company, but this was the one that put my GM over the edge)... their Tech told the customer that the cable company owned ALL the lines inside the TV can, and that he was going to have to disconnect one of their Dish receivers because we used one of their lines (this line ran right into the customer's ON-Q box in their 2 story home, mind you) and said they would have to call Dish to come redo the install "the right way"...
In other words, because we got their first, and he didn't want to run a line into the attic, he lied to the customer and deinstalled half of the customer's satellite system.
We "fixed" it and got the customer's system back up, but the cable company received a nice bill from Dish Network, which they eventually paid. I haven't seen this happen since then....