Why cathode is connected to the negative terminal instead of being positive?
In a galvanic (voltaic) cell, the anode is considered negative and the cathode is considered positive.
In a galvanic cell the reaction proceeds without an external potential helping it along. Since at the anode you have the oxidation reaction which produces electrons you get a build-up of negative charge in the course of the reaction until electrochemical equilibrium is reached. Thus the anode is negative.
At the cathode, on the other hand, you have the reduction reaction which consumes electrons (leaving behind positive (metal) ions at the electrode) and thus leads to a build-up of positive charge in the course of the reaction until electrochemical equilibrium is reached. Thus the cathode is positive.
In an electrolytic cell, the anode is taken to be positive while the cathode is negative.
Because In an electrolytic cell, the reaction proceeds in the presence of an external potential helping it along. The battery pumps electrons away from the anode (making it positive) and into the cathode (making it negative). Also, electrolytic cells are the reverse of a galvanic cell in the case of secondary (reversible) batteries.