The short answer is that yes, some of the same mathematical and physical techniques used to describe angular motion can also be applied to compactified dimensions, though they’re also measurable in length units like meters. Although the String Model does raise complications in the math, the extra dimensions themselves aren’t too bad, mathematically.
The long answer, I’m not qualified to give, and it probably wouldn’t fit in a message board post, anyway.
There are a couple of reasons we don’t observe individual gravitons. First, typical gravitational wave sources are extremely low frequency (the highest we expect to occur in the Universe are in the vicinity of kilohertz), and the individual particles corresponding to such frequencies would have ludicrously low energies. Second, gravity is so much weaker than the other forces that you would have to have a great many gravitons to get any interaction at all.
That said, it is possible that the reason that gravity is so weak compared to the other forces is related to the scale of the various dimensions. Under some models, if we could measure gravity at suitably small scales (small compared to the compactification scale of the extra dimensions), it would in fact be as strong as, say, the Weak Force. Such models don’t actually say how big the extra dimensions should be, though (it depends on how many of them there are), and there’s the possibility that they might be at experimentally-realizable scales, so there are groups doing experiments to measure gravity at smaller and smaller scales. I think they’re down to a tenth of a millimeter, now, with no detectable deviations from Newtonian gravity yet.
Don’t forget that gravity has no dipole moment, so you’re looking for a quadrupole right off the bat. That is, we can’t tell the difference between the lowest terms in the effects of gravity and our choice of coordinates.
As for the OP: no, the compactified dimensions are still as spatial as any other dimension. If they exist, we’d measure “distance” using centimeters (or a suitable fraction thereof) just the same as we do the first four dimensions.
The standard example is a garden hose. I can measure the distance along the hose (about 3000cm) and I can measure the distance around it (about 3cm). I don’t use an angle to describe position around the hose. In fact, if I used an angle I couldn’t tell the difference between a hose and a pipe, since using an angle throws away information about the radius.