To what extent does economics use Mathematical Logic?

As pointed out in the comments this question asks two different questions. I will try to answer them separately.

Question 1: To what extent does economics use Mathematical Logic?

Answer: while an 'extent' is not something that can be easily measured I think it is fair to say that in theoretical part of economics is virtually omnipresent. Consider the following fact:

MWG Microceonomic Theory - which is arguably the most popular and comprehensive graduate text for microeconomic theory has a rigorous proof derived from rigorous definitions and propositions on virtually every single page (save for preface, practice questions and reference page and few pages here and there). That is about 900 pages out of about 1000 page book that rely heavily on formal mathematical logic.

In different sub-fields of economics the importance of rigorous mathematical proofs can vary but in most of them I cant think of any graduate text that would not lean heavily on formal mathematical logic. Maybe philosophy of economics or history of economic thought would be some exceptions (and even there for example Philosophy of Economics: A Contemporary Introduction by Julian Reiss includes some parts that are written in formal mathematical logic)

Question 2: Do economics degrees require this [referring to subjects you mention inside the question] Formal Mathematical Logic?

Answer: Generally not - I could not find any survey on this issue but I think its fair to say majority of economic degrees do not have such courses that you mentioned as requirement to graduate and obtain a degree. The reason for that is that if you are good at mathematics you will be able to understand and even construct proofs without ever sitting in any course on formal mathematical logic. Any study of mathematics implicitly also teaches you formal mathematical logic - just not in necessary structured or obvious way.

As a matter of fact these fields [referring to the courses you mention] are relatively new subfields of mathematics and as Michael Greinecker pointed out in comments, "Mathematicians have proved theorems long before these subfields emerged." To see this you can just look at the beautiful and rigorous proofs in Euclid's Elements that were drafted thousands of years ago yet most of them still hold up and are quite rigorous.

This being said, taking formal class on introduction into making formal mathematical proofs should improve your own proof-making and understanding of proofs and help you to make your own future work more rigorous. The question is whether this benefit outweighs the opportunity cost of that lost time - but that is opinionated question I won't go into.

I think this can be the best explained by following analogy:

This is like asking to what extent economics uses linguistics and its sub-fields of grammar, stylistics etc. Well again linguistic is omnipresent in economics as most papers are not just proofs with zero text. However, is it necessary to take academic writing class or class in stylistic or grammar to be an economist or to get an economic degree? Well no, I think its fair to say most science degrees will have only optional writing classes and many not even that.

Many people will just learn how to write excellent academic works just by reading other papers. However, at the same time any person's writing would undoubtedly benefit by taking some academic writing class where you are exposed to stylistics and other sub-fields of linguistics even if it is not required to ever have such class.