1) The statement that general relativity (GR) is not renormalizable - is it a statement only about the quantization of GR or is it non-renormalizable also as a classical field theory?
2) More basically - is there a notion of renormalization group (RG) flow and renormalization vs. non-renormalization in classical field theory? I assume that the answer should be yes, since it seems to me that also a classical theory could have a range of validity (in energy / length scales) and that outside the range we would need additional terms or different values of constants in the Lagrangian. Also, we know that in statistical mechanics we speak about RG flow.
3) If the answer to question 2 is yes, then how is the running of constants formalized/calculated? Do we get different results of the running when using the classical vs the quantum theory of the field?
4) And if the answer to question 2 is no, then what is the fundamental difference between the quantum theory and the classical regarding to RG flow?