My math book gives 2 rules:
Rule 1: If $f(x)$ is increasing and $g(x)$ is decreasing (or vice versa), then equation $f(x) = g(x)$ either has no solutions or has only one solution.
Rule 2: If on some interval (x=a..b) $y=f(x)$ has maximum value $A$ and $y=g(x)$ has a minimum value $A$ (same) then equation $f(x) = g(x)$ is equivalent to the system of 2 equations: $f(x)=A$ and $g(x)=A$.
Question 1: for rule 1 what kind of monotonic function is meant - strictly monotonic (first derivative is never equal to zero) or weakly monotone (non-increasing/non-decreasing, meaning first derivative can be zero at one or several points).
I think that at least one function need to be strictly monotonic while the other is allowed to be weakly monotonic - its first derivative can be zero at multiple points. Or more presicely, for equation $f(x)- g(x) = 0$ to have only single (or none) solution function $y=f(x)-g(x)$ must be strictly monotonic (and it is achieved if at least one of them is strictly monotonic, the other can be strictly or weakly monotonic).
Question 2: for rule 2 - it can be applied to any functions f(x) and g(x) an there is no special requirement that either of them is monotonic (weakly or strictly) and there are no any other special requirements of any other nature.