(Microeconomics) How to classify Public Goods and Common Resources in Externalities?
First let's clarify some terminology: I assume that with "common resources" you mean goods that are non-excludable but rivalrous. The term that applies in those cases is open access. In common resources there are shared property rights so some excludability (this is one of the mistakes in Hardin's tragedy of the commons).

The answer to your two related questions then is: it depends on the type of good. Public goods and externalities are related but distinct problems.

Take a classic public good: national defense. That is not a problem where production or consumption have unintended effects and hence there are no externalities. It is also a very clear example of a public good: one cannot be excluded from national defense and one person's protection does not diminish someone else's protection.

That is not to say that the two never go together. Air pollution is a classic example of a production consumption externality (the companies do not intentionally pollute the air, but reduce consumer health nonetheless) and often a public bad: if one lives in the area one cannot be excluded and breathing the polluted air by one doesn't reduce the overall pollution level. 

The classic Gordon-Schaefer fishing model is both an open access resource problem and a production-production externality: the catch of an additional fish by one producer raises the costs of other producers.