Does flow of charge depend on electric potential or electric potential energy? Suppose I have a positively charged object A (+100C) and two other charged objects B (+10C) & C (+5C) in the vicinity of A i.e. in A's electric field. They are both equidistant from A. So, we can understand that B has a higher electric potential energy than C with respect to A due to the higher magnitude of charge of B.
Now, if I connect B and C by a wire, then positive charges will move from B to C in order to make their electric potential energies equal.
I understand this part.
Now, what will happen if we remove A? Now, we have nothing to compare B and C with, except each other. By comparing them with each other, we find that they both have the same potential energies with respect to each other. If we connect them by a wire, what will happen now?
Also, what if we make B negatively charged? What will happen then?
Also, does the flow of current depend on electric potential energy or electric potential?
 A: 1.We usually calculate the potential to infinity and not to a near charge,
so You can always compare C and B.
there is only the factor q between  potential and potential energy, so it makes no sense to differentiate between them. a charge q gains or looses its kinetic energy if it moves in a potential difference.
A: The short answer to your question is potential energy. Absolutely everything is always trying to get to the lowest potential energy state possible.
Let's start with the fundamental difference between electric potential and electric potential energy: electric potential energy is always in reference to a specific charge (the energy of the charge,) while electric potential is in reference to a region of space (the voltage across a battery.)
A bit of a problem here is that your allowing the amount of charge to change when connecting the two charges with a wire. In that scenario, to truly talk about the potential energy, you would have to add up the potential energy of each individual electron/proton. We can kind of do this in the case without charge A, it explains why all the charges get evenly spread: the charges want to be as far as possible from each other, so they all end up compromising and being equidistant from one another. However in the case of having charge A, talking about the electric potential energy of each individual charge is not practical. This is where electric potential comes in.
The electric potential describes what will happen to charges once they are put into that region of space. In this case, the potential from charge A will decrease inverse of the distance from charge A. Positive charge wants to get to lower electric potential (because that allows it to get to a lower potential energy.) This means the charge will want to get as far away as possible and you'll end up with 7.5 C of charge concentrated at each end of the wire, evenly spread by the same argument as above.
Conversely, negative charge wants to get to the highest electric potential possible, because that gets it to a lower potential energy. In this case by lower potential energy I mean a more negative potential energy. If charge B is -10 C, then you'll have partial cancelation with charge C and end up with -5 C that will be concentrated at the point closest to charge A on the wire.
This reveals why it is often better to talk about electric potential rather than electric potential energy in circuits. You don't want to worry about what each individual electron is doing, you just want an idea of what the charge will do overall.
A: Potential energy difference from 2 locations doesn't determine the flow of electrons. Locally this is true, macroscopically no. as e.g imagine a straight line path, From A to B then B to C . Imagine B is the midpoint, and from A to B the field points towards the center and then from B to C The field also points towards the center. The net potential difference would be zero( if constant E field) as the 2 contributions cancel. However clearly charges WILL move toward the center. The flow of charge is described by F = qE By definition E = -∇V so F = -q∇V This is the direction of flow of electrons
Or F = $- \nabla ( qV) $Here,$ \nabla( qV)$ points in the direction of increase of potential energy, so because its times by a minus, the force is in the direction of the IMMEDIATE decrease in potential energy. This Is a local law, so simply saying the potential  energy difference from point A to point B is negative so charges MUST move towards there, is wrong
A: Apparently your charged objects are metallic. Electrons tend to go to the lowest potential. However the potentials depend on the undefined shape of the objects, so the problem cannot be solved. I will then assume that each object has the same shape so that the capacities of all three of them are equal. In this case charge will tend to distribute equally between objects B and C, as long as A is at equal distance.
A: 
Now, what will happen if we remove A? Now, we have nothing to compare
B and C with, except each other.

Pick any point in space you want and assign it a potential of zero. Conventionally this would be a point at infinity. But in fact it is arbitrary. In circuit node analysis some convenient point in the circuit is assigned a value of zero. Then the potential at B and C is measured with respect to that point.
The fact that zero potential, or any other value of potential for that matter, can be arbitrarily assigned shows that potential is not a rigorous concept. It’s my understanding that Griffiths, in his book "Introduction to Electromagnetism",  makes the following statement:
"Evidently potential as such carries no real physical significance, for at any given point we can adjust its value at will by a suitable relocation of 0"
The only thing that really matters is the potential difference between two points.

If we connect them by a wire, what will happen now?

Nothing, since there is no potential difference.

Also, what if we make B negatively charged? What will happen then?

Current will flow since there will now be a potential difference.

Also, does the flow of current depend on electric potential energy or
electric potential?

It is the force of the electric field on the mobile electrons in the conductor that results in the flow of current. The electric field is, in turn, the gradient of the potential (volts per meter) along the length of the conductor.
Hope this helps.
A: if we make B negatively charged after removing A then the charges on B and C would be -38.33c and +38.33c respectively hence the electric field would be in the direction C to B and the flow electrons would be from B to C until both B and C end up becoming neutral.
electric potential energy causes the flow of electrons in a circuit, as its definition says "electric potential energy is the energy required to move a charge against the direction of electric field".
