The main thing you need to get your head around for this is that it isn't actually the neurotransmitter starting or stopping an action potential; it's just making it easier to do so. Let's take your example of dopamine: Dopamine can be either an excitatory or inhibitory CNS neurotransmitter. When it's released, it binds to one of its many receptors on the post-synaptic membrane, which in turn opens an ion channel associated with the receptor. If that ion channel is an Na+ channel, sodium will flood in and the membrane potential will be brought closer to the threshold for an action potential. Many of these will make the membrane reach threshold; thus an action potential will form. If the ion channel is for Cl-, then chloride ions will flood in instead, hyperpolarising the membrane and making it much harder for the membrane to depolarise. Each neurotransmitter has many subtypes of receptor, each of which is bound to a different ion channel (or G protein in some cases), which mediates the response given.

2 - not necessarily. As discussed above, it could be inhibitory and nothing could happen. Equally, say we're talking about an interneuron here - there's a sensory neuron on one end and a motor neuron on the other. If the sensory neuron wants to turn the motor neuron off, the interneuron needs to be turned on to turn off the motor neuron, so releasing the same neurotransmitter wouldn't necessarily be of any use... Sorry, a diagram would probably be easier for that. The sympathetic nervous system, for example, uses acetylcholine at the ganglion end, and releases norepinephrine at the target end.

3 - I have no idea. It would depend on the type of neuron as to which receptors it had, and hence which neurotransmitters it responds to in which way. I really have no clue and I don't know whether there even is a helpful answer to that.

4 - As far as I know, they're shared across all animals, particularly mammals, and some are found in plants.