No, 7 squared is 49 and 8 squared is 64.
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None. There is no perfect square that, when doubled, equals 60.
The square root of (any number that isn't a perfect square) is irrational.
Unless I've made a mistake, it's 3600 which is 60 * 60.
No real short cut I know of, just learn your multiplication tables and know your perfect squares. Once you do that you recognize that 36 is the perfect square of 6 so 3600 is the square of 60.
No, 325 is not a perfect square however 324 is a perfect square.