Do I understand electricity and the related phenomena correctly? Does a charge have an electric field only when put in relation with another charge? Is E potential determined based on position relative to the charge?
E feilds have no intrinsic value. They are only relevant when put in a relation between two charges. They appear to be an abstraction used to rationalize the rules followed by particles that somehow have a mysterious property called charge.
The position within this field is what determines electric potential which is a measure of potential energy per unit of charge that gets converted in to work as a result of EMF.
Is this coherent?
 A: You might think of the electric field as a collection of vectors (with one for each point in space) representing the Coulomb force that would be applied to a charged particle, dependent on its charge, at each point (https://en.wikipedia.org/wiki/Electric_field). Thus, the electric field exists regardless of charges. If there are not any charged particles, the value everywhere is 0. It is only physically manifest, however, if there are charged particles to apply forces to one another. 
As far as electric potential, you are right in an electrostatic sense. There exists a potential between two points based on their relative positions in the field (https://en.wikipedia.org/wiki/Electric_potential).
However, electromotive force has to do with circuits. It is the voltage across the terminals of the battery (or other voltage source) in a circuit.
I don't know if this is exactly what you were looking for, but I hope it provides some perspective.
A: 
Does a charge have an electric field only when put in relation with another charge?

If you want to discuss this on a philosophical level you'll end with the tree in the forest which will make or not a sound then falling over. Without observer no proof.
But there is a good reason to believe - as long as there are no contradicting evidences - that the electric field of a charge is an intrinsic (ever existing) property. Imagine a  surface with an elastic glue on it and stick your finger on this glue. Withdraw your finger and until you reached some distance the funger feels an force. This is a good analogy to what should happen to an electron in an interaction with a proton. During an approach the interaction should happens instantaneous at some small distance and during the removal the force between them shoul last longer. This observation was not done and that's why the electric field is called an intrinsic property of charges.

Is E potential determined based on position relative to the charge?

From the above said it is clear that the electric potential of an isolated charge at any point in space has always the same value. Having two or more charges reduces (for opposite charges) or rises the potential at any point (in relation to an isolated charge). BTW this is important for the interactions between the protons in the nucleus of an atom as well as for the electrons around the nucleus.

E fields have no intrinsic value. They are only relevant when put in a relation between two charges. They appear to be an abstraction used to rationalize the rules followed by particles that somehow have a mysterious property called charge.

If an electric field of a charge in an interaction with another charge wouldn't have well defined values we couldn't use equations to describe this interactions. But we can precisely and this statement of your is not well founded.

The position within this field is what determines electric potential ...

Correct.

which is a measure of potential energy per unit of charge that gets converted in to work as a result of EMF.
Is this coherent?

Sorry, but I'm not really know what you want to say.
