Are inhibitory synapses governed by different chemicals than excitatory synapses? If a neuron has both excitatory and inhibitory chemical synapses providing it with input, is it true in all organisms that the chemicals that cause the inhibition are distinct from those that cause the excitation?

Related question:
What is &quot;synaptic polarity&quot; in a chemical synapse?
Short answer
I cannot give an exclusive answer, but if you are talking about a schoolbook neuron with different neurons synapsing onto it, then excitation is indeed generally mediated by another neurotransmitter than inhibition. Why?...

Background  


First and foremost. While recognizing that neurotransmitters in themselves are not excitatory or inhibitory, it is the receptors that govern their workings, it is a fact that in the central nervous system the principal excitatory neurotransmitter is glutamate, and the principal inhibitory neurotransmitter is GABA. In fact, none of the Glu receptors (ionotropic or metabotropic) afaik is inhibitory, and none of the GABA receptors is excitatory. 
Secondly, from a postsynaptic point of view, synapses with different input cells are separate. In other words, a glutamatergic neuron never secretes GABA in the same synapse or vice versa. In effect, neurotransmitter systems are separated. Often, a principal neurotransmitter may be accompanied by neuromodulators, but never will one neuron release opposite effector molecules. Instead, opposing inputs are imposed by separate cells, each with their own neurotransmitter system.
The neurotransmitter acetylcholine can be excitatory at the neuromuscular junction in skeletal muscle, causing the muscle to contract. In contrast, it is inhibitory in the heart, where it slows heart rate. So Ach can have dualistic and even opposing actions, but the respective receptors are different and geographically segregated in different neurons;  
But, this is schoolbook, anthropocentric knowledge, and if you ask any creature, anywhere in the multiverse, then yeah... I don't know.