The coin-like thing on the table is a marker used to indicate the player who is the dealer, and this marker is called the button or dealer button. The button circulates around the table in a clock-wise direction one player at a time after each completed game. Now in casino games, the player holding the button does not actually act as the dealer and deals the cards to each other players, instead an employee from the house does it. Since the player with this position is always the last to act thus giving the player an advantage, hence the rotating button is there to prevent cheating so that every player gets to take turns serving as dealer.
As for the terms you mentioned given to the players:
- Cowboys is pocket kings(KK), a reference to a starting hand e.g. I got the cowboys
- Pro should be self-explanatory
- Shark describes a player who's very good/expert
- Big dog means a big underdog to win the pot
Hope this helps