Well, sir, it is difficult to answer your question about the lowest height or shallowest aquarium in which seahorses would thrive because that depends largely on the type of seahorses you will be keeping and how large they grow. For instance, the diminutive dwarf seahorses (Hippocampus zosterae), which are no bigger than your thumbnail when fully grown, typically do very well in small aquariums ranging from 2-10 gallons, which may be no more than 10 inches tall. On the other hand, the giant breeds such as Hippocampus abdominalis required tanks at least 30 inches tall in order to mate comfortably and successfully accomplish the transfer of eggs during mating.
But generally speaking, I would not recommend keeping any of the commonly kept seahorses (e.g., Hippocampus erectus, H. reidi, H. kuda, H. barbouri, H. comes, H. angustus, H. subelongatus, etc.) in aquariums that are not at least 20 inches tall, and tanks that are at least 24 inches tall are preferable for these species. To give you an idea of why this is important, Mic Payne, a professional aquaculturist who operates the Seahorse Sanctuary, found that maintaining a population of Hippocampus subelongatus in shallow tanks only 16-inches (40 cm) deep resulted in chronic problems with various forms of gas bubble disease.
For best results, I would look for an aquarium no lower than 20 inches high for keeping medium-sized seahorses such as H. fuscus, Tim.
Best of luck finding the perfect ponies and just the right aquarium for your needs and interests, sir.