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As for your first question, yes you are correct. If there are more unknowns than equations in a homogeneous system, then there will be infinitely many solutions.
However, we don't know what happens for the other cases. The statement doesn't mention anything about them. For example, you can have systems of equations with the number of equations and unknowns equal to each other and have infinitely, none, or exactly one solution. We don't know until we actually solve it.