How do we know that gravity is spacetime and not a field on spacetime? How do we know that gravity is the curvature of spacetime as opposed to a field, which couples equally to all objects, on spacetime?
 A: Practically speaking, what's the difference?  There exists a rank-two tensor field on spacetime called the "metric" $g_{\mu \nu}$ which couples to all mass-energy, and things that we intuitively call "gravity" happen when that field deviates from the Minkowski metric $\eta_{\mu \nu}$.  Whether you want to call the metric "spacetime itself" or "a field on spacetime" is basically just terminology.
Steve Weinberg, very unusually, likes to think of general relativity as being just another field theory, with the metric as just another field, and dislikes the "geometric interpretation" that is usually taught, where we describe the metric as being spacetime.
Note, however, that certain stress-energy tensors are only compatible with certain spacetime topologies, so you can't just think of every spacetime as just being $\mathbb{R}^4$ with a funky metric on it.
A: 
How do we know that gravity is spacetime and not a field on spacetime?

We don't because it isn't. I should explain that spacetime is an abstract mathematical space, not real space. The Earth is surrounded by space, not spacetime. There is no motion through spacetime because it models space at all times. See relativist Ben Crowell saying that here. A gravitational field however is not an abstract thing. It's a region of space where light curves and your  pencil falls down. This gravitational field is often described as curved spacetime. However spacetime curvature is associated with the tidal force, which is the second derivative of potential. Your pencil doesn't fall down because of tidal force. It falls down because of gravitational force, which is the first derivative of potential. This is why you will never find Einstein saying light curves because spacetime is curved. Instead you find him saying this:
"This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that 'empty space' in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials g$_{\mu\nu}$), has, I think, finally disposed of the view that space is physically empty".
There's perhaps some confusion as regards spacetime and space here, but IMHO the crucial point is that a gravitational field is a place where space is "neither homogeneous nor isotropic". As a result the speed of light is spatially variable, which is why light curves and your pencil falls down. Because the state of space in front of your face is not quite the same as the state of space down by the floor. You can find Einstein reiterating this in 1929. He was talking about electromagnetic fields and gravitational fields, and he said this: 
"The two types of field are causally linked in this theory, but still not fused to an identity. It can, however, scarcely be imagined that empty space has conditions or states of two essentially different kinds, and it is natural to suspect that this only appears to be so because the structure of the physical continuum is not completely described by the Riemannian metric".
Again there's perhaps some confusion as regards spacetime and space here, but IMHO what stands out is that a gravitational field is a place where space has a particular state. Space is inhomogeneous where a gravitational field is, such that if you plot the inhomogeneity, your plot is curved.  

How do we know that gravity is the curvature of spacetime as opposed to a field, which couples equally to all objects, on spacetime?

It isn't quite either of above. It's a field, but not "on spacetime" per se. A gravitational field is a place where space has been conditioned or altered by the presence of a concentration of energy typically in the guise of a massive planet, this effect diminishing with distance in a non-linear fashion. This is then modelled as curved spacetime. That's where your metric, your plot of measurements made with rods and clocks, is not uniform. It is instead inhomogeneous, in a non-linear fashion. Clocks go slower when they're lower, but there isn't a linear relationship between height and rate. So your metric is curved. However space is not curved, just as water is not curved under the sea where sonar waves curve:
 
A: It was believed that gravity only affects things with mass.  But then Einstein predicted that light would bend around the sun slightly, which was famously proven correct during a solar eclipse.  So gravity seems to affect light which has no mass.  This was a problem, except that Einstein already had a solution, which was that gravity warps spacetime itself.  This allowed photons to behave as though they were being affected by gravity, because although they are actually moving in straight lines unaffected by gravity, they are following straight lines in curved space (which happens to be curved by gravity).
Afaik, if gravity is just a field, then there needs to be some explanation as to how gravity affects things that don't have mass.  Furthermore, the standard model predicts that the Higgs boson is responsible for matter having mass, and gravity.  So we would then need TWO fields to explain gravity.  One field mediated by Higgs bosons which is responsible for matter interacting with gravity.  And then we'd need another field mediated by an unknown particle which would be responsible for non-massive (mass-less) particles (like photons) interacting with gravity.
A: The reason for regarding gravity specifically as being spacetime, as opposed to any other field, is related to inertia.
Inertia is motion that takes place without forces. So in Newtonian mechanics, inertial motion is moving in a straight line at constant velocity. Mass is the constant of proportionality between the force exerted and the acceleration of the object on which the force acts.
General relativity was motivated in part by Einstein's lift thought experiment. If you're standing in a box being held in a gravitational field, you experience a force opposing the direction of the field as a result of not moving. That is, the floor of the box exerts a force up on your feet when the gravitational field would move you toward the floor. Since gravitational and inertial mass are the same, this is indistinguishable from inertial motion in an accelerating lift. There are some complications like tidal effects, but you can reduce those so they are arbitrarily small by considering a small enough region compared to changes in the gravitational field.
The lift experiment explains that it is physically impossible to distinguish between gravity and inertia. If you attribute inertia to spacetime, then the gravitational field is spacetime, or so the argument goes.
