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I wouldn't worry about the request for a cardiology clearance, as long as you don't have TB or are HIV positive, then there's no medical reason that the UAE government should deny you entrance into the country. I can't speak for the medical requirements of your hiring company though.
Otherwise, if it's an oil company, their HR department is most likely very experienced in dealing with work permit issues and navigating the channels you have to go throught to get an employment pass.
For more information on this process though, have a look at the Visas to Abu Dhabi page.