The abstract equality and inequality operators (== and !=) convert their operands if the operand types do not match. This type coercion is a common source of confusion about the results of these operators, in particular, these operators aren't always transitive as one would expect.
In the statement false B, both the operands are strings ("" and "0"), hence there will be no type conversion and since "" and "0" are not the same value, "" == "0" is false as expected.
One way to eliminate unexpected behavior here is making sure that you always compare operands of the same type. For example, if you want the results of numerical comparison use explicit conversion:
Or, if you want string comparison:
Side-note: Number("0") and new Number("0") isn't the same thing! While the former performs a type conversion, the latter will create a new object. Objects are compared by reference and not by value which explains the results below.
Finally, you have the option to use strict equality and inequality operators which will not perform any implicit type conversions.