It means that you have 7 total cards... The 59 is the total you "could eventually get"... But for now you have 7. That is good, its a start and something we can build on ;) ...
The summoners are the cards that are like the leaders of the team. They are the ones that decide who plays and who doesn't and they do this by "summoning" the monsters to the team. In reality it is you that makes this decision, but you will find the right summoner to match the team you want to play. (for instance, if you want the fire team, then you will have to pick the red summoner to get the players from that color)
The gold cards are the same cards, but they have a gold look to them and they are more rare. The reason for this is to have a collectors edition, plus they can be used to play in "gold card only" tournaments.
And yes I think that if you saw the game that I have on this page, then you are going to be perfect! This game is more about getting your strategy right in picking your team, and then the computer will play it based on your choices. So definitely you computer and internet connection are no factors! You are good :)