How do we actually visualize the effect of two fields interacting in the same region of space?
If fields are just mathematical formulations to explain things that have no physical meaning, how are field interactions formulated and studied theoretically that can explain practical observations?
The concept of fields is the concept of assigning numbers to every instance of time and to every point in space. Think of temperature. You can assign a temperature value to every point in space whenever you like by going there and measuring the temperature with your favourite temperature probe. Now consider another field: wind speed. This is a vector field, i.e. it consists of 3 numbers corresponding to direction and magnitude. Clearly, both fields interact since wind can carry away heat energy or transport colder air to a given place. You can now start to reason about whether or not the temperature and wind fields are real or in fact made up by different entities like particles (or again fields). What is irrefutable is the description of a physical phenomenon, wind and temperature, and their interactions. Whether this is the most fundamental one, you can never know.
Good question.
Almost all practical and scientific progress in high energy physics in the last decades has been related to lattice regularization of field theories.
(If you have access to it, see Kogut's article)
Not only has this direction brought about genuine accurate predictions in high energy physics but it provides an intuitive connection to statistical mechanics via the Feynman path integral. In this framework fields live on sites, links, etc... of a space-time lattice. Interactions can be seen with your eyes plain as day in terms like: $\phi^{\dagger}_{x} U_{x, \mu} \phi_{x+\mu}$, which shows an interaction between two fields on neighboring sites ($\phi_{x}$ and $\phi_{x+\mu}$) via a gauge field on the connecting link ($U_{x,\mu}$).
I would say if you are interested in making contact with reality in field theory to look into ``lattice field theory'' and start reading papers from the late '70s.
As for whether such a model has ``no physical meaning'', this is a question of philosophy, and you can ask it on SE philosophy, but keep in mind the practical quality of holding a belief that your successful model is actually a reflection of reality: you gain intuition about how the natural world interacts and behaves and as such you may be able to even improve your understanding and make progress in explaining un-explained phenomena.
First you suggest fields are somewhere specific, raising the question of how they can share locations. Then you suggest fields don't describe real physics, raising the question of how they can exhibit cause and effect. Neither way of thinking about fields is right. They're functions, and each has a value everywhere. Just as $x^2$ doesn't have a location, nor does $A_\mu$; just as $f^2+g^2=1,\,f=g'$ constrain two functions, so field equations constrain the functions we call fields.
