I'm surprised that none of the people more familiar with GR have brought up Kaluza-Klein theory so far. It's not an answer to the OP's questions (which have to do with much simpler physics), but it's an absolutely beautiful piece of work, highly interesting, and is at least what I immediately think of when the question "are gravity and [electro]magnetism the same?" is asked. So, despite it being tangential, and 3 in the morning here, I think it's interesting enough to warrant a post
And possibly a thread hijack.
Einstein came up with his field equation in 1915, which in mathematical language both stunningly simple and unimaginably complex describes how the curvature of spacetime (on one side of the equation) is proportional to a given distribution of mass and energy (on the other side). A second equation takes in that spacetime curvature and lets you compute the paths that one takes in that spacetime, and voila - we're able to describe the effects of gravity in general relativity.
Now, these equations don't depend on the number of dimensions of spacetime - it's usually trivial to solve the Einstein field equation in as many dimensions as you'd like. So some fellows (including the named Kaluza and Klein) decided to imagine that there was an extra dimension of space, and solve Einstein's equations in this new 5-dimensional paradigm, with one caveat - that the laws of physics don't depend in any way on that fifth dimension (a question in itself). As an added bonus, they did so by setting the matter/energy side of the equation to 0 - in other words, taking Einstein's equation in a 5-dimensional vacuum.
A beautiful thing happens - by separating the 5-D Einstein equation into its components, we can come up with three very interesting, smaller equations. One describes something called a scalar field (don't ask!). Another is simply Einstein's equations in 4 dimensions, but right side of the equation is no longer 0 - rather, it describes a field of electromagnetic radiation! The third equation is perhaps the most shocking - nothing more or less than Maxwell's equations (condensed into one equation) describing the electromagnetic force.
So from higher-dimensional gravity, electricity and magnetism "appear" in 4 dimensions, and from a higher-dimensional vacuum, we get "stuff" - radiation! Light! Fantastic little result, and in many ways the basis of the higher-dimensional theories (such as string theory) that we have today. Higher dimensions seems to be useful for getting other forces out of gravity.
There are ways to extend this mechanism to include other forces, like the strong and weak force, and this has been tried with varying levels of success. There hasn't been a solution for the other forces quite as elegant and beautiful as the Kaluza-Klein solution, however, and correspondingly it's been difficult to make other particles besides photons appear from a higher-dimensional vacuum. Regardless, it is clear that there's more to gravity than meets the eye, and if there's but one higher dimension, hidden from our view, magnetism and electricity are, indeed, the same as gravity.