Your ship will have "Satellite" TV. One cruise I was out on our balcony, 89 degrees & off the coast of Acapulco Mexico, sipping an "Adult Beverage" and my wife started laughing. There was a very bad snow storm viewing on our TV. The Satelite TV hub must have been out of Denver Co. They were showing the city of Denver in a bad blizzard. To answer your question, yes you can view most of the major TV networks.
We usually watched CNN.
Your question regarding the computers, I'm not sure. When we're on a cruise, we're on vacation...no cell phones and or computers.
As for your question about Grand Cayman Island... You can go to the shore ahead of your tour group. In this port you will be tendered, (they will use the Island's tender boats/they are larger and carry much more passengers). Most all of the ships tours will meet up at or very near the customs building, (this place is nuts/crowed very small place and the tour group will hold up signs showing just which tour group). When you get to the shore, walk to the right along the shore to a set of buildings. Great place for shopping, (yellow building as well as other buildings). Also make sure that you visit the town of "Hell" , the "Tortuga Rum Cake Facrory" and last but not least the "turtle Farm". I'll attach some pictures. Have a great cruise...We leave in 7 days on the Carnival Glory. Dave & Diane.