Quantum mechanics and quantum mechanical theories are totally independent of the classical ones. The classical theories may appear and often do appear as limits of the quantum theories. This is the case of all "textbook" theories - because the classical limit was known before the full quantum theory, and the quantum theory was actually "guessed" by adding the hats to the classical one. In a category of cases, the full quantum theory may be "reverse engineered" from the classical limit.
However, one must realize that this situation is just an artifact of the history of physics on Earth and it is not generally true. There are classical theories that can't be quantized - e.g. field theories with gauge anomalies - and there are quantum theories that have no classical limits - e.g. the six-dimensional $(2,0)$ superconformal field theory in the unbroken phase. Moreover, it's typical that the quantum versions of classical theories lead to new ordering ambiguities (the identity of all $O(\hbar^k)$ terms in the Hamiltonian is undetermined by the classical limit in which all choices of this form vanish, anyway), divergences, and new parameters and renormalization of them that have to be applied.
Also, the predictions of quantum mechanics don't need any classical crutches. Quantum mechanics works independently of its classical limits, and the classical behavior may be deduced from quantum mechanics and nothing else in the required limit. Historically, people discussed quantum mechanics as a tool to describe the microscopic world only, assuming that the large objects followed the classical logic. The Copenhagen folks divided the world in these two subworlds, in an ad hoc way, and that simplified their reasoning because they didn't need to study quantum physics of the macroscopic measurement devices etc.
But these days, we fully understand the actual physical mechanism - decoherence - that is responsible for the emergence of the classical logic in the right limits. Because of decoherence, which is a mechanism that only depends on the rules of quantum mechqnics, we know that quantum mechanics applies to small as well as large objects, to all objects in the world, and the classical behavior is an approximate consequence, an emergent law.
To know the evolution in time, one needs to know the Hamiltonian - or something equivalent that determines the dynamics. The previous sentence is true both in classical physics and quantum mechanics, for similar reasons, but independently. If a classical theory is a limit of a quantum theory, it of course also means that its classical Hamiltonian may be derived as a limit of the quantum Hamiltonian. Of course, if you don't know the Hamiltonian operator, you won't be able to determine the dynamics and evolution with time. Guessing the quantum Hamiltonian from its classical limit is one frequent, but in no way "universally inevitable", way to find a quantum Hamiltonian of a quantum theory.