Factors refer to whole numbers. The square root of 820 is not a whole number.
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A Square Root is one of the two equal factors of a number.
No there is not. If you are looking for prime factors of a number and you get to the square root of that number you can stop. Yes, there is. If an integer is not itself a prime, then one of its factors will be less or equal to its square root and the "co-factor" will be greater than or equal to the square root. But both cannot be greater than the square root so, when searching for factors, you can stop when you reach the square root.
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The square root
If a number has three factors, it's a perfect square. One of those three factors (apart from 1 and the number itself) would be the square root.