Does electricity pass through synapse? As far as I understand, when a neuron fires the action potential generated in a cell body reaches only the presynaptic part of the synapse, then triggers the transmitter to pass through the gap, and that transmitter causes the membrane in the postsynaptic part to open ion channels, pumps, etc ..., which generate a completely new electric potential that finally reaches the receiver cell.

So the electricity generated in a firing neuron never actually reaches receivers (not directly)?
Yes – electricity does pass through synapses if the link between the neurons in question is constituted by an electrical synapse.

There are two different types of synapses, chemical synapses and electrical synapses.

At a chemical synapse, an arriving pre-synaptic action potential indeed causes the release of neurotransmitters which carry the signal across the synaptic cleft to the subsynaptic membrane. This can effect an action potential in the postsynaptic neuron (among other things).

At an electrical synapse, on the other hand, ions do actually move from the presynaptic cell, through a gap junction channel, and to the postsynaptic cell, capable to cause a depolarization there.

Electrical synapses are found in many animals (including human). Due to their different characteristics, like the lack of neurotransmitter-related mechanisms and a significantly smaller synaptic cleft, they e. g. allow for a faster propagation of a signal between two neurons (while the signal decreases in strength due to the lack of gain). They occur as a part of neural structures which require fast signal transmission.

(While more technical details can be found in the referenced sources, it is helpful to review some of the basic concepts to avoid misconceptions:

It is sometimes (naively) assumed that the propagation of an action potential consists in a flow of electric charge along the axon, but what actually happens is that ions move through the membrane, i. e. roughly along an axis that is orthogonal to the direction of the signal. It is the action potential that moves along the axon. Thus, the original question is not to be taken to mean »Does the electric current that moves along the length of the axon also move through the synaptic cleft?«

Still – even if the flow of electric charge does not constitute the action potential, electricity can flow along the axon, as well. This is what happens in the case of saltatory conduction between the nodes of Ranvier in myelinated axons. It is generally referred to as an electrotonic potential.)