I think I am misunderstanding Einstein's equivalence principle and his elevator I'm having difficulties understanding why a gravitational acceleration can be guaranteed to be locally equivalent to an accelerating frame. Doesn't it matter on how the force is being applied? If the floor of the elevator is exerting a force on me (due to some external force accelerating it) then this would be very different from a gravitational acceleration that would accelerate each part of my body equally. If I held a string up during my acceleration in the elevator and I let go, the string should fall, while in the gravitational case, it would remain the same, or am I missing something?
 A: 
If the floor of the elevator is exerting a force on me (due to some external force accelerating it) then this would be very different from a gravitational acceleration that would accelerate each part of my body equally.

No, it wouldn't. The two situations are experimentally indistinguishable. That's one of the points of this thought experiment.
An even more important point (at least to me) is that it highlights a problem with the Newtonian concept of an inertial frame. Suppose a mad scientist alien teleports you to an elevator car. You're weightless. The alien tells you that you might be orbiting a star, or you might be in one of those huge voids in space. You need to use the local physics experiments piled up in a corner of the elevator car to determine which situation applies. Can you do it? (The answer is no.)
Next the alien teleports you to another elevator car in which you feel Earth normal gravity. The alien now tells you that you might be stationary on the surface of a non-rotating planet, or you might be in a spaceship in otherwise empty space accelerating linearly at 9.8 m/s2. Once again, you need to use the local physics experiments to determine which situation applies. Can you do it? (The answer is once again no.)
Newtonian mechanics tells us that there's a big difference between the orbiting elevator car and the elevator car in a void between galaxies. The first is non-inertial, the second is inertial. But there's no way to distinguish between the two using local experiments. General relativity is consistent with the experimental results; it says that both situations constitute local inertial frames.
Newtonian mechanics also tells us that there's a big difference between the elevator car sitting on the surface of a planet and the elevator car in an accelerating spacecraft.  The first is inertial, the second is non-inertial. Once again there's no way to distinguish between the two using local experiments. General relativity is once again consistent with the experimental results; it says that in these cases, both situations constitute non-inertial frames.
A: The answer is very simple and you may forget everything about lifts, elevators and so on and so forth.
The gravitational force (which does exist, I wonder the comments above) is the only interaction in the universe where the dynamics does not depend on the mass of the particle (the so called statement that inertial mass is equivalent to gravitational mass), because the masses on both sides cancel out. To start with, assume you are in a gravitational field $\textbf{G}(x)$: your equation of motion is
$$
m\ddot{\textbf{x}} = m\textbf{G}(\textbf{x}) 
$$
namely $\ddot{\textbf{x}} = \textbf{G}(\textbf{x})$. Assume now that you are instead in an accelerated system where the non-inertial contributions are exactly equal to $\textbf{G}(\textbf{x})$, with no other forces acting. The equation of motion will then be
$$
m\ddot{\textbf{x}} = m \textbf{G}(\textbf{x}) + \textrm{real forces} = m \textbf{G}(\textbf{x})
$$
which is identical to the one you have in the gravitation field of magnitude $ \textbf{G}(\textbf{x})$. Now if you just look at the dynamics can you tell whether you are in a non-inertial reference frame or in a gravitational field? No, you cannot. Hence the equivalence.
A: I'm having difficulties understanding why a gravitational acceleration can be guaranteed to be locally equivalent to an accelerating frame.
Actually, it can't. See section 20 of Relativity: the Special and General Theory where Einstein said this: 
“We might also think that, regardless of the kind of gravitational field which may be present, we could always choose another reference-body such that no gravitational field exists with reference to it. This is by no means true for all gravitational fields, but only for those of quite special form. It is, for instance, impossible to choose a body of reference such that, as judged from it, the gravitational field of the earth (in its entirety) vanishes”. 
You can't transform away a real gravitational field. So the room you’re in is not exactly equivalent to the room in the rocket. See this article and how it refers to an infinitesimal region? That's a "local" region of zero size. That's no region at all. The principle of equivalence was "Einstein's happiest thought", but it's just a principle, it doesn't mean being in a real gravitational field is exactly the same as being in an accelerating rocket.     
Doesn't it matter on how the force is being applied? If the floor of the elevator is exerting a force on me (due to some external force accelerating it) then this would be very different from a gravitational acceleration that would accelerate each part of my body equally.
It's different, but generally speaking, you can't tell the difference between the floor pressing up on you or you pressing down on the floor. Like David Hammen said, in practice you can't distinguish between the two using local experiments. Gennaro Tedesco said much the same re if you just look at the dynamics But note what CuriousOne said: you wouldn't be able to tell, to first order. If you had super-precise measuring equipment, such as NIST optical clocks, you could tell. Especially if your room had a high ceiling.    
If I held a string up during my acceleration in the elevator and I let go, the string should fall, while in the gravitational case, it would remain the same, or am I missing something?
I think you're missing something I'm afraid. In both cases the string falls down. Or maybe I'm missing something! 
