They can be either, but not together.
y = x and y = -x are both monotonic.
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No. For example, y = 7 is monotonic. It may be a degenerate case, but that does not disallow it. It is not a bijection unless the domain and range are sets with cardinality 1. Even a function that is strictly monotonic need not be a bijection. For example, y = sqrt(x) is strictly monotonic [increasing] for all non-negative x. But it is not a bijection from the set of real numbers to the set of real numbers because it is not defined for negative x.
Negative * positive = negative Positive * positive = positive Negative * negative = positive
Negative * positive = negative Positive * positive = positive Negative * negative = positive
A negative divided by a positive is negative. A negative divided by a negative is positive. A positive divided by a positive is positive. A positive divided by a negative is negative.
positive and a positive is a positive negative and a negative is a positive to answer your question: positive and a negative is a negative.