'The Middle Ages' is capitalized because it is the name of a specific period in European history; you will (almost) always find it capitalized, even when it is used to refer to other geographical areas as in 'Japan's Middle Ages'. 'Government' you should not find capitalized unless it is used to refer to it as a sort of personified entity, as in 'Government has joined with Big Business to squeeze out the small entrepreneurs'.
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government always lc, including British government, except in official titles such as Government House and Her Majesty's Government.... If in doubt use lower case.