Is potential energy a type of energy at all? Is potential energy, whether it be that of a charge in an electric field or a mass in a gravitational field or anything like that, actually an energy that the particle itself contains, like kinetic energy? Or is it just a measure of its ability to do work? 
Is it the case that instead of integrating conservative forces over distances to find work done, we use the fact that the work done by a conservative force doesn't depend on path and hence we just use the notion of 'potential energy' and it's variation with distance, and just take the difference between the potential energy at two points to easily find the work done? And hence, is potential energy nothing but a tool to calculate work done by conservative forces?
 A: Maybe we can say that a particle "contains" chemical energy, since we have to break its chemical bonds to access to chemical energy, and maybe we can say that a particle "contains" nuclear energy, since we have to break nucleons to access to it.
Kinetic energy, however, depends on movement, which means that it depends on the frame of reference of the observer. So for sure we cannot say that a particle "contains" kinetic energy. Think for example about a ball sitting still in a moving train: how much kinetic energy does it have from the point of view of someone standing on the platform? And for someone sitting inside the train?
In a similar way, potential energy depends on the position of the particle relative to something. If we move the particle, its potential energy is generally going to change. So we cannot say that a particle "contains" potential energy either.
However, this doesn't mean that kinetic or potential energy are not "real", since we can always use them to perform work on or to heat something, which it's what really matters in the end.
A: Well, as the name suggests, its the potential of the body to store energy inside it or some other way (whatever you like), potential energy is actually potential to do work. But having a constant potential energy means nothing.
as F = - Del (V)
This means , its something , not nothing.
Just think of Gravitational potential energy (stored in a body), it will come down.
A: A body's kinetic energy is the work it can do because of its motion (as it comes to rest). Calculate this amount of work and, in Newtonian physics, you find it to be equal to $\frac{1}{2}mv^2$.
A body's potential energy at point P is the work it can do by changing its position (in a conservative field) from P to another point, O, which has been chosen by convention as the point of zero potential energy.
So capacity to do work is common to both KE and PE. I'm not sure that PE is any more 'just a tool to calculate work' than KE is. 
I do agree, though, that we think of KE as residing in the moving body, whereas PE does not reside in a particular body; it's quite useful to think of it as residing in the field.
A: 
Is potential energy, whether it be that of a charge in an electric field or a mass in a gravitational field or anything like that, actually an energy that the particle itself contains like kinetic energy?

Yes. If you lift an electron, you do work on it. You add energy to it. You increase its mass. Then when you drop it, some of that mass-energy, which we call potential energy, is converted into kinetic energy. After this kinetic energy is dissipated, you're left with a mass deficit. See the Wikipedia binding energy article. The potential energy was extra mass energy that was converted into kinetic energy as the electron fell. Gravity didn't add any energy, it just converted one form of energy into another.   

Or is it just a measure of its ability to do work?

It isn't just that. It's real energy. When you separate two masses you do work on them. As a result their mass increases in line with E=mc². You can get the work back from them when they fall back together.     

Is it the case that instead of integrating conservative forces over distances to find work done, we use the fact that the work done by a conservative force doesn't depend on path and hence we just use the notion of 'potential energy' and it's variation with distance, and just take the difference between the potential energy at two points to easily find the work done? And hence, is potential energy nothing but a tool to calculate work done by conservative forces?

No. It's a very real thing. It applies to the electron and the proton in the hydrogen atom, and to the bullet and the sled. But note that whilst p=mv means momentum is shared equally, KE=½mv² means the less massive body gets more of the energy. 
