While Buddhists do not believe in a God, they certainly believe in supra-physical states like the jhanas and nibbana and supra-material processes like rebirth and kamma.
While I cannot claim expertise on the subject, no I do not think that these terms: Rebirth and Kamma mean to them what you think they think that they mean to them. Mahayana is called 'The Great Vehicle'. Its trappings have the appearance of superstition, however the Southern schools and the Zen schools more openly discuss that the Buddha is not superstitious but aims towards skepticism. I do think that some versions of Buddhism are superstitious but the more superstitious varieties appear designed to invade, to propagate and to replace war with peace. Look at Japan.
Japan is originally quite serious about gods, war, honor, ancestors and such; but then Buddhism appears and begins to influence the culture. It moves people towards a philosophical way of life, and that appears to be by design. Around 600CE in the Warring Period, Japan gets introduced to Mahayana Buddhism and gradually becomes less war-like over time. Around 1100CE Toyatoma Hideyoshi convinces the Satayama clan to surrender to a lord named Nobunaga, and subsequently unites Japan under Nobunaga. The land forgets its Samurai who become Ronin and spend their times studying Buddhism. Buddhism preaches harming no one. It is thought that the Samurai become Ronin because the land becomes unified, their jobs mostly redundant. The possibility of unification, however seems attributable to Buddhism's gradual increasing influence, which uses all available tools including Taoism and superstitions to influence the culture. War loses its popularity. Sure it is just guesswork coming from me. I think its an example of religion slowly pacifying a war obsessed region.
Also many Buddhists believe in deities and Bodhisattva-s. So it does not seem to be a problem to say Buddhist practices and laws are also based on a supramundane worldview. Also religion always had a distinctly divine overtone
Sure they do, but I think they are taught to question whether their own beliefs are real. I would have to check with someone more familiar with the particular school.