How do academics in humanities such as philosophy apply mathematics or physics in their discipline?

First, a historical note: The idea that philosophy and mathematics are separate fields is a recent invention. A few hundred years ago, they were spoken of as one and the same. If you go back even further to the 16th century, physics becomes part of that same field, and even further back the term “philosophy” referred to any systemic investigation whose result was knowledge. For example, Newton’s great work was titled Principles of Natural Philosophy (in Latin) and while Aristotle more famously wrote on ethics and metaphysics, he also wrote on zoology, logic, physics, and biological taxonomy amongst other topics. Herodotus (primarily a historian by modern standards) and Hippocrates (a primarily a physician) were both considered philosophers. Herodotus wrote important religious theory in his history books and Hippocrates wrote important ethical ideas in his medical works (ask any western doctor about the “Hippocratic Oath”). These didn’t seem like digressions and interludes to contemporary readers. These were all works addressing topics of philosophy.

1. What is the difference between the Philosophy of Computer Science and Computer Science?

Roughly, science studies X and the philosophy of science studies people who study X, ideas about X, and the connection between X and other philosophical questions. Topics in the philosophy of X (for every science) include questions that define the scope and nature of the field, as well as its epistemic standards such as research methods and standards of evidence.

Three examples of questions along these lines are:

1. In physics, there is generally a preference for a theory that possesses “mathematical elegance” over one that appears to be more ad hoc, even when the two theories have identical observed evidence. Is having this preference legitimate? If so, can it be justified based on the practice of physics alone or does it require appeal to other principles (maybe ontological principles, or epistemic ones)?
2. In computer science, one studies computers and computation. Given an object, how do you tell if it’s a computer? What does it mean to solve something “via a computation” or “via an algorithm” as opposed to via other means?
3. In physics, one studies the physical properties of objects. Someone might be tempted to say that mathematics analogously studies the mathematical properties of objects. However, we are used to thinking of physical properties as being those embodied by physical objects that exists in physical space (that is, mass is a property of this ball that exists in the world). Is there an analogous sense in which mathematical properties are embodied by mathematical objects that exist in mathematical space?

Over the course of the investigation of these topics, topic-specific questions tend to crop up. Contemporary philosophy of mathematics is extremely concerned with ontological and epistemological questions, while contemporary philosophy of medicine is primarily concerned with ethical issues and the philosophy of biology involves questions about the precise nature of life and the implications of extraterrestrial life.

It’s quite possible that someone could be an expert in subjects like this and not be able to do cutting edge mathematics and physics, though answering these questions certainly requires a substantial knowledge of the fields of science you are analyzing.

2. Are there well-publicized research performed by religious scholars or philosophers that actually inform or advances or guides the development of mainstream theory in say astrophysics, biology, medicine, mathematics, etc.?

The answer historically is yes, even setting aside the historical note I opened with. Thales, Aristotle, Alhazen, Ptolemy the Second, and Jean Buridan are all famous people who would be considered philosophers by today’s standards who made substantial contributions to physics, astronomy, biology, medicine, mathematics, and other fields of science. If you’re looking for more recent contributions it’s more difficult.

There seems to be a historical bias in categorizing people as scientists when they did both science and philosophy (by modern standards). Although he’s far more famous for his work on calculus, Leibniz was also a very influential philosopher, and probably more influential as a philosopher in his time than as a mathematician. You’ve probably heard of his theory that we live in the “best of all possible worlds,” because an all powerful and all good god wouldn’t let us live in anything else. These ideas were so influential in his day that Voltaire became famous in part for attacking Leibniz’s philosophy, such as in his book Candide. Einstein, Minkowski, Turing, and Nash all did philosophy, but that aspect of their work is less well-known to the general public.

There are some people who are widely classified as philosophers who made substantial scientific advances in recent times. The work of the logician Gottlob Frege (1848 - 1925) was ignored in mathematics for a while, but the mathematicians Bertrand Russell and Giuseppe Peano would revolutionize mathematics by introducing axiomatic systems of logic and arithmetic grounded in Frege’s work. Linguistics is an example of a science that recently grew out of a field of philosophy, and so many recent philosophers have had substantial impacts on linguistics, such as Ludwig Wittgenstein (1889 - 1951) and Saul Kripke (1940 - Present). The philosopher of mind and philosopher of language David Chalmers (1966 - Present) is hugely influential in the development of fields of computer science such as artificial intelligence and human-computer interaction. Chalmers is one of those people whose ideas are used to benchmark and understand developments in AI, as he has written extensively on the concept of intelligence and coined the term “the hard problem of consciousness.”

David Hume (1711 - 1776) wrote a book called an Enquiry Concerning Human Understanding which is undoubtedly one of the 10 most influential philosophical works ever written in the West. Although it’s impact is most prominent in philosophy (Kant famously claimed that it woke him from his “dogmatic slumber”), it also was highly relevant to scientists and the sciences. Einstein had the following to say about it in a letter to Moritz Schlick, 14 December 1915

You have also correctly seen that this trend of thought [positivism] was of great influence on my efforts, and specifically E. Mach and still much more Hume, whose treatise on understanding I studied with fervor and admiration shortly before the discovery of the theory of relativity. It is very well possible that without these philosophical studies I would not have arrived at the solution.

This wasn’t one of your main points, but I also wanted to address this:

ultimately I am not certain about how subjects such as mathematics and physics can be made relevant to philosophy (or other areas).

They don’t need to be “made relevant to philosophy.” Anything that increases ones understanding of the world is something that is relevant to philosophy. However, one could wonder to what extent the technical details of scientific theories or scientific methods affect philosophy. Like any question about the interdisciplinary use of techniques developed in one field but applied to others, the answer to highly depends on the field of philosophy and the skill of the philosopher, but virtually every scientific advance gets used by some philosophers for something. There are ontologists whose arguments about the philosophy of time that revolve around technical details of Einstein’s theories of relativity, epistemologists who use category theory, philosophers of mind who use results from cutting edge neuroscience research, and ethicists and metaethicists who read and make extensive use of results from social sciences such as economics, behavioral psychology, and decision theory.

As to your second question,

Are there well-publicized research performed by religious scholars or philosophers that actually inform or advances or guides the development of mainstream theory in say astrophysics, biology, medicine, mathematics, etc.?

I can think of two examples.

Ethics is a branch of philosophy, and it certainly informs and guides the development of theory in science. It furnishes some first principles about how to do science, that are relevant across disciplines. It is also additionally relevant in specific fields like medicine and biology.

Besides influencing how research is conducted, it also is important for determining what is studied and why. For example, if medical research has the goal of saving lives, where did that goal come from? It has a moral/ethical basis.

I'm sure you will find plenty of philosophical and religious research that informs science in this regard (though I'm not an expert and can't easily provide some examples).

It might not be quite what you had in mind, but I do think it satisfies your statement "inform[s] or advances or guides the development of mainstream theory".

Also religion and medicine have some overlap in that they both (often) share a goal of caring for people. Religious thought has some influence in medicine for this reason. For example, chaplains usually have an official role in hospitals, even hospitals that have no religious affiliation.

Religious thought similarly influences some branches of medical research for this reason. There is a healthcare research area in providing "holistic patient care" (meaning caring for all of a patient's needs, not just medical--not to be confused with "holistic medicine" which is typically a term for quackery). Religious thought has some influence in this area.

Again, I'm not an expert, but here is one example of an article on the influence of religion in palliative (end of life) care. Note that it's published in a medical journal, and its authors include a doctor and two religious leaders.