Is the term $$(\partial^{\mu}\phi)^{\dagger}$$ same as $$\partial^{\mu}\phi^{\dagger}$$ for any complex scalar field $$\phi$$?
It depends on what inner product you are using to define the $$\dagger$$ operation. If you are including integrals over $$x$$ as part of the inner product, then $$(\partial^\mu)^\dagger = - \partial^\mu$$,just as it is with the momentum operator in QM. If $$\phi$$ is an operator on a Hilbert space and you are only taking the adjoint on that space, then the $$\partial^\mu$$ is unaffected by $$\dagger$$. I expect that the latter is what you means as it looks like something you would find in the action for a scalar field.
As always, $$\dagger$$ needs a specification of the inner product to define it.