How is charge generated by the action potential is distributed to all of the neuron connections? From what I understand the total charge transmitted by a neuron once it fires is same for every neuron. Is it divided based on synaptic weights?
The action potential is an active process, its mediated by voltage sensitive ion channels that shuffle ions around in order to create an electrical gradient that travels through the whole cell. When it reaches an axon, some of these positive ions bind to pockets of molecules (vescicles) that fuse with the cell membrane to release their neurotransmitters. Check out this very good answer to a similar question. Once these neurotransmitters bind to receptors in the post-synaptic membrane of the subsequent cell it produces an EPSP.
Each synapse is unique, meaning that although the action potential is the same synapses are changed depending on their activity some synapses might have more vescicles, more neurotransmitter, more sensitivity to the change in voltage. Postsynaptic membranes can also grow and increase the density of receptors in order to make the synapse more effective. The effectiveness or weight of a connection is just a measure of how good that connection is at eliciting a spike or action potential from the postsynaptic cell.
Once a cell fires it fires, but not all the connections count the same to elicit a response from the next cell down and there are a bunch of interesting sub-threshold dynamics.