Can Eve break this public key cryptosystem if she can solve DLP or DHP?
Can Eve break it if she can solve Diffie-Hellman problem?


Yes, at least, the computational Diffie-Hellman problem.  This problem is "given the triplet $h, h^x, h^y$, recover $h^{xy}$"

Let us assume that Eve has an Oracle that solves this problem.  Then, she sets $h=m^{ab}$, $h^x = m^b = (m^{ab})^{a^{-1}}$ and $h^y = m^a = (m^{ab})^{b^{-1}}$, and passes $h, h^x, h^y$ to her Oracle.  The Oracle returns with $h^{xy} = (m^{ab})^{a^{-1}b^{-1}} = m$, problem solved.

And, if she can break the Discrete Log problem, she can solve the CDH problem, so this answers that question as well.

N.B. The protocol you listed is known as the "Shamir Three Pass Protocol", and is currently viewed more as a curiosity rather than something that solves some problem better than other solutions.