A nerve cell with dopamine receptors gets an action potential and releases dopamine to other neurons. Does this nerve cell only release to cells with dopamine inside? Because what if a neuron has a dopamine receptor but serotonin as a transmitter?
No, neurons with all sorts of neurotransmitters frequently synapse onto neurons that release other types of neurotransmitters. In fact, aside from the most common neurotransmitters like glutamate and GABA, this is the rule rather than the exception. You mention specifically dopamine, and this is a perfect example--the medium spiny neurons in the striatum that are the most "famous" dopamine-receiving cells are NOT dopaminergic themselves--they are all GABAergic. The other aminergic neurons are similar; the noradrenergic locus ceruleus neurons and the serotonergic neurons of the raphe nuclei project to much of the forebrain, which contains few noradrenergic or serotonergic neurons of its own.
Neurons frequently have receptors for neurotransmitters that they do not produce, and these are commonly known as heteroreceptors, as opposed to autoreceptors, which respond to the same transmitter that the cell possessing the receptor responds to. Autoreceptors commonly exist more for the purpose of negative feedback ("I've already released enough of my transmitter, therefore I need to stop so I don't overload the cells I synapse onto") than for neurons releasing the same transmitter to activate one another.