Air isn't a natural A sound. CARE-et is just the way North Americans say it. Its a lot less pronounced in other countries and more true to its spelling.
As far as I know ( haven't been in school for a long time so brace yourself ) the letter A is only interpreted as either A ( like day ) or ah ( like blah ). Different words make it sound differently in North America because North Americans have weird ways of saying a lot of words.
If card and carrot have different pronunciations to you its entirely because of the way you're used to saying the words and has nothing to do with the core language rules themselves. Then again language rules for English are almost useless given the number of contradictions which makes it among the ( if not THE ) hardest second language to learn in the world.