Are all fields in the universe we know of quantum fields? Are all fields in the universe we know of quantum fields?
Do all fields that exist must be inherently quantum in nature?
How about fields that are yet to be discovered (ie. a new field like Higgs field) , do they all have to be quantum fields?
 A: Currently all fundamental fields are quantum, except for gravity. For this reason Quantum Gravity is a hot area of research, but the full Quantum Gravity theory has not been developed yet. Why not?
The challenge is not just technical, but conceptual. On one hand, the Quantum Field Theory cannot consistently co-exist with any classical theory. If the Quantum Field Theory is correct, then gravity must be quantum. On the other hand, gravity cannot be just another quantum field theory, because gravity bends the space and time ("the background"), on which the Quantum Field Theory is based, and this creates unreasonable challenges (time is steady and independent in QFT, but depends on the field and is dynamic in GR) that technically result in non-renormalizability  of quantization.
The only logical way to resolve this contradiction is to admit that both theories, General Relativity and Quantum Field Theory, are  approximations of another unknown yet theory that in itself is neither General Relativity nor Quantum Field Theory. As mentioned in the comment of @Prahar above, chances are that gravity will be quantized in some way, but Quantum Gravity will not be a standard Quantum Field Theory.
Other possibilities also have not been ruled out, such as that gravity may not have a quantum nature or have a nature that would change our understanding of "quantum" and what exactly we mean by it. Thus the answer to your question is that no one knows yet.
A: We believe that fundamentally the universe is quantum mechanical. So we would expect all fundamental fields to be quantum in nature. However that does not mean that all fields are quantum. 
The laws of physics depend are scale-dependent (Joseph Conlon gives a very good exposition on this in Why String Theory?). Thus the nature of the field will depend on the scale of the theory/problem. If the scale of the problem lies in the classical domain then the fields of interest will be classical in nature. In principle we can solve such a problem with quantum fields, but in practice it does not quite work that way.  
