If brane theory says that gravity is weak because it extends into another spatial dimension, is it also saying that the gravity attributed to dark matter might be due to matter in another spatial dimension?
None that I have read about. There are a LOT of string theories out, many have branes; that is two dimensional strings.
In addition
string theory does offer possible candidates for dark matter since since many theories include a LOT more particles and forces than has been observed (detected). String theory seems to offer no specific predictions as to which of many possible candidates are the dark energy or dark matter
As far as my layman's research took me (I had a very similar idea), the Randall-Sundrum-Modell specifically mentions this effect as a possible source for dark matter.
I also thought that, depending of "our" brane's shape in a super-space, our very own matter might be just as much of a candidate as well, though that sounds even wilder (and, in my case, has to be called philosophical thinking, since I have no educational basis in physics - just math and IT).
One well-known but still incomplete idea is that string theory reduces to a low-energy effective SUSY theory which could be similar to the MSSM. Then the standard SUSY mechanisms (e.g. neutralinos, ... as DM candidates) apply. Unfortunately neither a realistic reduction to the SM or MSSM is known, nor is the generation of small but nonvanishing masses understood.
In these models DM located on a second brane is not required.
None that I have read about. There are a LOT of string theories out, many have branes; that is two dimensional strings.
Just to be careful, branes are not 2D strings. Branes are nonperturbative objects that arise as solutions to the supergravity equations of motion. They can be in any number of dimensions (up to the total dimensionality of the theory), although some string theories can only have odd or even dimension branes. The proper notation is a Dp-brane, where p denotes the number of spatial dimensions. Most popular brane world models posit our universe to be a D3-brane, with 3 uncompactified spatial dimensions.
Now, you can have a D1-brane, which is 1-dimensional like a fundamental string, and you can have higher dimensional Dp-branes wrapped around p-1 compactified extra dimensions, leaving only 1 extended dimension, also like a string.
