Your back would have been more insulated because it was up against the chair, it's perfectly normal.
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But it doesn't just do this when I get out of a chair. If i walk for a long distance, it starts sweating. Sometimes even happens if I am standing still.
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Armpits don't really sweat that much and you always feel like your back sweats more because it has a larger surface area. Do you usually put on several layers/carry a backpack around? If so that could be why as well.
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Sometimes, but this happens even when I don't. Actually, it seems to sweat more when I don't.