2. If you are referring to the property in algebra that $\frac{a}{b}=\frac{c}{d}$ implies $ad=bc$ provided of course b,d are nonzero... then yes, this is a valid and completely sound property of real numbers.
Perhaps what your professor means by "illogical" is that cross-multiplication is not a fundamental property or operation in algebra. The correct and logical step is to multiply both sides of the equation by $bd$. The term "cross multiply" is not so much a mathematical tool as it is a teaching tool, to help young (particularly math-hating) students remember how to proceed when faced with this type of problem.