Does quantum mechanics require classical mechanics for its own formulation? Is true that quantum mechanics requires classical mechanics (as a limiting case) for its own formulation?
 A: I really don't understand the question, but I seldom do. QM   "corrects" classical mechanics, stretching it like relativity, and includes it as a tricky "classical limit", but it crucially further includes deeply novel features, like the statistical nature of the theory, and the uncertainty principle.
Contrary to the original breathtaking "reinterpretation formulation" of Heisenberg (Umdeutung) in Hilbert space, which bests reflects noncommutativity, the much subsequent phase-space formulation (alias "deformation quantization") keeps much of the language of classical mechanics almost intact, and extends some algebraic rules and casts  its formal rituals in a statistical language. So it looks like a more direct stretch of classical mechanics, thinkably the path classical mechanics was stretched on other planets, far, far, away,  a long time ago, which did not have the benefit of Heisenbergs and Schroedingers and Diracs.
Hard to tell whether that planet was luckier than ours or not...
A: Newtonian mechanics works extremely successfully for describing motion on a huge range of scales. Quantum mechanics has to be able to replicate those successes in the regime where Newtonian mechanics is known to work well.
A: There are different theories that we call classical theories. There is Newtonian mechanics with all its later developments. Then there is classical field theory, such as Maxwell's equations for electromagnetism. And then there is thermodynamics, with statistical physics. Quantum mechanics is built on concepts from all these classical theories. So, without the different classical theories that inform quantum mechanics, we would not have been able to formulate quantum mechanics.
BTW, that is also why the idea that classical theories can be considered as a certain limiting case of quantum mechanics is somewhat misleading. Which classical theory should we end up with in this limit?
A: Suppose a voltage input is supplied to a system, then its output is similar to input after some time. If input is step, then output is step and if input is pulse then output is pulse, that is staedy state response. But what is going in-between till output is stablise. There is transient response in-between input and staedy state. Quantum mechanics is an attempt to find what happens between cause and final classical result. In classical mechanics, applying pressure or giving energy to a body cause motion (i am not writing change in motion), and quantum mechanics wants to know how that energy is distributed among constituents of given body. It tries to satisfy our curiosity of how final state is reached. For doing so, it use old technique of divide and rule, reductionalism or atomism, dividing a body into fundamental particles, atom. So it is basically theory about how atom behave. So in this way it is quantized (into minimum unit), discrete (only integer value is allowed because no further divison) and probabilistic as wave (we don't kow what is going in-between but only result) like nature that diminishes particle like property of locality.
To illustrate it further we take very first example of quantum mechanics. It is microstate entropy given by Boltzmann as traeting system composed of number of particles or molecules as famous Boltzmann law of entropy. Sum of all microstates must equal to macrostate, so entropy of all particles equal to entropy of system. For that suppose a some amount of heat, $Q$ is given to a system having ideal gas at constant temperature, $T$. If no work is done, then all heat is converted into increasing internal energy by $E$. Then the entropy of the system is given by,$$S=\frac{Q}{T}=\frac{E}{T}=Nk$$This is equivalent to raising $N$ particles from ground state to energy level, $E$ per unit of temperature. Now entropy of $i^{th}$ particle for above energy and temperature condition is given by Boltzmann law as,$$s_i=k\ln W\implies\sum_{i=1}^Ns_i=S$$$$\begin{align}S&=\sum_{i=1}^Nk\ln\frac{N}{n_i}=k\sum_{i=1}\ln N-k\sum_{i=1}^N\ln n_i\\&=kN\ln N-k\sum_{i=1}^N\ln{N\exp{\frac{-E_i}{kT}}}\\&=kN\ln N-kN\ln N+k\sum_{i=1}^N\frac{E_i}{kT}\\\sum_{i=1}^Ns_i&=\frac{E}{T}=\frac{Q}{T}=S\end{align}$$So this is proved that total sum of quantum effects results to classical result or sum of transients leads to steady state, in other words classical is sum value with no statistical variation left as all included.
