My questions are the following:

1. I saw that a field was defined as a correspondence of a vector or a scalar to every point in space. I asked my teacher about this definition, and he told me that there is a better one: “a field is a region in space, where a particle with a certain property (such as being charged or having a mass) would feel a force acting upon it. My problem with his definition is the fact that he ignored the fact that fields are vector or scalar quantities, not a “region in space” – a region in space has no magnitude and direction. Is his definition is correct? Later I read another definition – “a field is an influence an object has on the space around it”. Is there a way to make that definition more precise and rigorous? In my opinion, there is a problem with the word “influence” – this word is not defined in a precise and clear way – it can be misinterpreted.
2. I read that the field is the one who is exerting a force on an object, not the object itself. I will try to be clearer – a charge creates a field around itself, and another charge wonders in that field. We then say that the force on the second charge is exerted by the field, not the charge that created the field. Could you explain this? It doesn’t make any sense – the field is just a model we created to describe the influence an object has on its environment. Is it just a convention that the field is the one who is exerting the force, not the charge who created the field?
3. How do the charges “sense” each other? Is it because of the field or the fact that they have the property of “being charged”?

Your three questions are three different ways of recasting a single problem: what is a field?

Let me start with what a field is not. It cannot be "a region of space" equipped with special properties. The reason is already evident in your comment, but I would add that there is no way to make sense of a negative or even complex region of space. There are historical reasons for this definition. Maxwell used this kind of definition to introduce the concept of field. But it should be seen as a pictorial way to speak about a physical property that shares with space a continuum nature. Unfortunately, this region of space definition is with us, and I am afraid it will remain, even if it is complete nonsense.

Now, what is a physical field? Stated this way, it looks like a tough question. It is longtime Physics is not more concerned about the intrinsic nature of things. For the excellent reason that investigating how things work has been demonstrated much more fruitful than asking what things are.

To understand the concept of physical field, one has to remember the conceptual steps which brought about the status of physical system for this quantity.

The starting point is the description of force, in classical mechanics, as a force at distance. This description looks familiar to us, but it is not free from conceptual difficulties. How a thing here knows about the presence of another thing there? The classical definition of field is based on the finding that in some cases, the force on a test body which does not perturb too much the system depends only on a function of the point. This allows a formal shift from an action at distance concept to a local interaction between a body and a local quantity (the field). This description, alternative but equivalent to the action at distance, looks just as a formal trick at the static level. However, when time variations are taken into account, one discovers that some fields seem to have their own dynamics. It is possible to attach to them physical properties such as energy, momentum, and angular momentum. In other words that these auxiliary quantities for describing forces behave exactly like other entities that we consider physical systems. At this point, the conceptual shift is to assign the status of physical system also the field. We do not know what an electron or a proton is, but we know that it is something carrying some properties we can measure. In the same way, we can assign similar properties to an electromagnetic field, whatever its intrinsic nature is.

At this point, saying that charges sense each other by the intermediate of another physical system, which is the local field, looks more fundamental than speaking about action at distance. The key point is that we are always speaking about our concepts to describe the world. We do not know what the world is. Still, this pragmatic approach works very well.

Summarizing, the concept of field is a useful concept to describe interactions in terms of quantities at the same point. It is convenient, but at the classical physics level, it would always be possible to maintain its auxiliary nature and remain with interaction at distance. The price would be an extreme complexity of the resulting formalism. In particular, a relativistic description of interacting bodies without the concept of field would be significantly more complicated, not to mention the problem of the transition to Quantum Mechanics.