How and in what way do force fields work and exist? 
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*Do fields constantly exist with their sources, for example, will a positive charge keep exerting a positive electric field irrespective of whether we put a small positive or negative charge nearby/within the field? 

*What if the field is generated instantly when something which can interact with it is immersed in it? How do we know that the field exists before we put an object into it?


This might come off as a silly doubt because I lack knowledge about what constitutes a force field and how forces are communicated through their fundamental particles.
You could be of great help if you could explain that to me. Thanks!
 A: A charge $q$ at a point $\vec r_0$ engenders an electric field,
$$\vec E = \frac{q}{4\pi\epsilon_0 |\vec r - \vec r_0|^3}(\vec r-\vec r_0).$$
As you can see, this field as written exists at all points. Now, if we have multiple charges $q_i$ placed at positions $\vec r_i$, then at a point $\vec r$, the net or total field experienced is,
$$\vec E_{\mathrm{net}} = \sum_i \frac{q_i}{4\pi\epsilon_0 |\vec r - \vec r_i|^3}(\vec r-\vec r_i).$$
A test charge $q_{\mathrm{test}}$ will experience a force $q_{\mathrm{test}}\vec E_{\mathrm{net}}(\vec r)$ at the point $\vec r$. While the field $\vec E_{\mathrm{net}}$ may look very different from a simple field pointing radially outward, the field due to each of the individual charges remains the same; they each contribute the standard field of a point charge.
As for establishing the existence of a field, we perceive them precisely because of interactions with them, such as on charges or mass, or in the case of quantum field theory, particles are viewed as excitations of fields and thus their existence supports the existence of the field.
A: Nice question! I will answer your question in the context of Classical Electrodynamics. Maybe someone else can elucidate on the QFT side of the issue.
Fields exist even if there is no test-charge. 
This makes the first part of your second question irrelevant because the charge doesn't need to know anything instantly to create a field at the point where the test-charge is put because the field would have already been there. 
The second part of your second question is how do we know if the field was there before we put the test-charge there. Well, there are two ways to look at the issue. 
The first is analyzing the issue with the full acceptance of the machinery of Physics that has been so-far developed as experimental facts - after all, it has been verified experimentally beyond any doubt. This is to say that the Maxwell's equations clearly tell us that the field exists at a point even if the test-charge doesn't and there has never been reported any violation of Maxwell's equations ever. So, we have no way but to accept the ontology that Maxwell's equations has, i.e., there are fields at a point even if the test-charges are absent there and these fields react upon the test-charge instantly as the test-charge is put there. 
The other way is of being more theoretical and asking why did we formulate a theory in a way that it has fields that exist even when the test-particle isn't there. It could have been the case that Maxwell just got his equations as a guess and they turned out to work. In such a case as well, since they work, we better accept what they say. But since we know Maxwell's equations were derived over a long course theoretical brainstorming, it is also valid to ask what were the theoretical motivations behind formulating the equations in the term of fields that exist even if the test-charges don't. And such kind of understanding is crucial (more often than not) for research in the relevant area. So, there are two strong theoretical motivations (not necessarily historically important but those that people consider to be important theoretically today) behind this field formulation of classical electrodynamics: 


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*Locality: Special Relativity teaches us that no information can travel faster than light. Thus, we better not have our theory require us to send the information of a test-charge being put somewhere instantly to the source-charge. But the test-charge does feel a force instantaneously when put around a source-charge. This means some very physical influence of the source-charge had to be present where the test-charge was put even before the test-charge was put there. 

*Energy and Momentum Conservation: The experimentally derived laws of interaction of charges dictate that the energy and momentum must be carried by the fields if the energy and momentum have to be conserved. This ascribes a perfect physical reality to fields. And if the fields have momentum and energy then they better not appear or disappear depending on whether we put a test-charge somewhere or not. 

*Electromagnetic Waves The existence of electromagnetic waves, the oscillations of pure electromagnetic fields in a perfect vacuum (i.e., in the absence of any test-charges) makes it inevitable to ascribe a test-charge independent physical reality to the fields. The existence of electromagnetic waves is, of course, a bare experimental fact. 
Along with these major theoretical motivations, it is also a major reason behind us having a field theoretic description of Classical Electrodynamics that all the attempts based on action-at-a-distance approach to the same have failed terribly. These failures are just consequences of the fundamental theoretical underpinnings stated above. 
A: First of all you should be clear in your mind about the meaning of "Physics" as a discipline. Physics is the study of nature with mathematical tools, creating mathematical models that can fit measurements/data and can successfully predict what new measurements/data will show. 
Measurements are real numbers, recorded from the behavior of matter ( particles, radiation, solid bodies ....) which are postulated in physics, matter exists and questioning its existence goes into metaphysics and philosophy. Postulates are what axioms are in a rigorous mathematical theory. The pick up the subset of mathematical functions useful for the study of nature.
With this as background, one immediately sees that Force is one of the postulates of Newtonian mechanics .  The mathematical description of force fields, as the example in Jamal's answer is a successful application of Newtons model, it allows to predict the behavior of charged particles with great accuracy ( above the level where quantum mechanics is necessary). So the concept of fields is a mathematical concept which used in calculations gives real numbers to be compared with the experiment/measurement.
One should not confuse  the mathematics , demanding reality from it, with the measurements. The mathematical model is as real as the measurements that support it. To ask for reality of force fields is metaphysics, on the platonic ideal road, and not physics.
With this in mind, 

Do fields constantly exist with their sources, for example, will a positive charge keep exerting a positive electric field irrespective of whether we put a small positive or negative charge nearby/within the field? 

In the mathematical models we use fields exist with their sources, and as Jamal's example  shows the electric fields combine if there is more than one source.
But it is dependent on the success of predicting interactions of charged particles using this model. The field is a useful mathematical icon.

What if the field is generated instantly when something which can interact with it is immersed in it? How do we know that the field exists before we put an object into it?

In the classical picture of electricity and magnetism, it is derivable from the postulates that the field exists before it is tested , "instantly"
In the quantum mechanical frame though, which is the underlying modeling frame of nature, ( all classical theories emerge from it), there is no "instantaneous" action at a distance, because the "force field" is carried by photons which are limited by the velocity of light. This last depends on special relativity and its postulates and the postulates of quantum mechanics which model nature with quantum fields.
One just has to keep in mind that the model is not the reality but a mathematical fit to the numbers measured for the given problem, the appropriate model has to be used for the appropriate phenomena observed or predicted.
