Are higher order logics substantially stronger than second order Do we get much by using logics of order higher that 2? Does each transition to next level provides much power?

There is a precise sense in which third-order and higher-order logics, with full semantics, are no stronger than second-order logic with full semantics. For any formula of higher-order logic $\phi$ there is another formula of second-order logic $\phi'$ such that $\phi$ is valid in full higher-order semantics if and only if $\phi'$ is valid in full second-order semantics. This is described in more detail in the Stanford Encyclopedia of Philosophy article on higher-order logic. It is a standard fact for any course about second-order logic.