What's the definition of distance in curved space-time in general relativity?

The spatial distance between two points is, if you pick a foliation by spacelike hypersurfaces, the distance between point $A$ and $B$ is the integral along the shortest geodesic linking the two points with that induced metric. In other words,

$$d(A,B) = \min_i \left(\int_A^B \mathrm d\lambda \sqrt{h^{ab} \frac{\mathrm dx_i^a(\lambda)}{\mathrm d\lambda} \frac{\mathrm dx_i^b(\lambda)}{\mathrm d\lambda}}\right)$$

Where $h$ is the induced metric on the hypersurface, and $x_i$ obeys

$$\ddot x^a + {\Gamma^a}_{bc} \dot x^b \dot x^c = 0$$

and goes through points $A$ and $B$. If you are unable to get a foliation, I'm not quite sure if the spatial distance between two points is well defined, globally at least.

If you want to do it for a particular observer, it's probably a good idea to pick Fermi coordinates with the proper time of the observer for the foliation.

The answer to your first question is:

The distance in general relativity is given by;

$ ds^2 = g_{\mu\nu} dx^\mu dx^\nu $

In special relativity (minkowski spacetime) this reduces to;

$ds^2= dt^2 -dx^2-dy^2-dz^2$ $ = \eta_{\mu\nu} dx^\mu dx^\nu $

If you know the metric of spacetime (obtained by solving Einstein's field equations) then you can find the distance between 2 events using the above formulae

Hope that helps! :D