Some questions on observables in QM 1-In QM every observable is described mathematically by a linear Hermitian operator. Does that mean every Hermitian linear operator can represent an observable?
2-What are the criteria to say whether some quantity can be considered as an observable or not?
3-An observable is represented by an operator via a recipe called quantization if it has an analog in classical mechanics. If not, such as spin since it has no classical analog, do we use data from experiment to guess what this operator could look like? are there any other methods for finding that?
4-Are there other observables, besides spin, which also have no classical analog?
 A: Questions 1,2:
An observable is an element that is obtained from experiments. You can take this as the definition of an observable. The fact that we make an operator and give it some properties does not change/influence the outcome of an experiment. It just so happens that the theory we have ascribes linear, hermitian operators to explain experiments. With this in mind, it is easy to say that not all linear, hermitian operators we cook up describe observables.
Question 3
Initially, the classical-quantum correspondence was used, but people quickly realized that it was of limited use. The modern view is that nature can be described by Group Theory (especially the Poincare Group) and everything that is observed follows from there. With this in mind, you don't have to guess about the existence of the Spin Operator, it comes up naturally. What is more important though, is the representations of the operator. When you relate theory and experiments, remember that you are dealing with the representations of an operator. An operator cannot be measured and is useless by itself unless you specify the basis.
Question 4 
I don't know the answer to this, but I can tell you that we never measure spin by itself, but the interaction of a spin with something else. Why? In my view, that is the definition of a measurement. 
A: In principle, every Hermition operator can be an observable in the sense this term is used in quantum mechanics. For finite systems (edit: i.e., those with a finite-dimensional Hilbert space) I have seen theoretical results that prove measurability of any Hermitian operators, though I couldn't locate a reference.
But measuring an observable becomes very difficult when it is a contrived operator rather than one of those you find typically discussed.  
A: The parity operator, the unitary operator which implements reflections on the wavefunction is both next to impossible to measure, and has no classical analog. This operator is unitary, but its real and imaginary parts are Hermitian.
The spin is not nonclassical, is the spinning particle. It's not like parity and other discrete symmetry observables.
