Why do cables and batteries have a positive and a negative side?
Electrons flow out one side (the negative one) and come back in from the other (the positive one). Current is not associated with electron accumulation, but with electron flow. The point of the battery is pushing electrons from the positive to the negative terminal: this pushing requires energy, that is chemically kept in the battery, used to push the electrons that then release it when they go through your circuit.

The negative sight might be or not ground. Ground is a word that is used to refer to a circuit particular node, sometimes connected to ground. You measure all voltages with respect to this node, so it is the zero voltage node. I understand you might think that electrons might escape if ground is connected to the negative terminal of the battery but that is not possible since they must go back in the battery because no net charge accumulation is allowed. If an electron pushed by the battery happens to escape through ground, another one will come from ground, make it through your circuit and then back in the battery.