What is the length of null geodesic? There are many questions about this but none of them adresses my concrete question. If it is indeed true that for light we have $ds^2 = 0$ does that mean that in 4d spacetime total "distance" is zero for light? By distance I mean lenght of a geodesic that light moves on? Describes? 
 A: 
If it is indeed true that for light we have $ds^2=0$ does that mean that in 4d spacetime total "distance" is zero for light?

Yes, but the scare quotes on the word “distance” are exceptionally important. The spacetime interval is not just a distance in spacetime. 
In space a distance is always just measured by a ruler. In spacetime the interval comes in three flavors, spacelike which is measured by rulers, timelike which is measured by clocks, and lightlike which is zero. 
Also, if two points in space are separated by 0 distance then they are the same point. The same is not true of the interval. There are many events with a spacetime interval of zero from any given event, they form what is called the light cone. 
Because the interval is zero between lightlike separated events we need a different parameter for identifying events on a lightlike line. This is the affine parameter. For timelike lines the affine parameter is proportional to proper time, but it also works for lightlike lines. 
A: The length of a null curve is indeed zero, as can be easily computed by hand from the metric tensor with the world function : 
$$s = \int_\gamma \sqrt{\operatorname{sgn}(g(\gamma', \gamma')) g(\gamma', \gamma')}$$
In the case of a null curve, $g(\gamma', \gamma') = 0$ for all points of the curve, so this isn't particularly hard to show. 
This isn't to say that notions of the "length" of a null curve do not exist. For a variety of applications, it can be useful to have some notion of distance along a null curve. The obvious candidate is simply the parametrization of the curve : for a curve $\gamma(\lambda)$, the distance from $p_1 = \gamma(\lambda_1)$ to $p_2 = \gamma(\lambda_2)$ is simply $\lambda_2 - \lambda_1$. Simple, but also completely arbitrary : you can simply rescale the parametrization via 
$$\gamma(\lambda) \to \gamma(f(\lambda))$$
with $f$ some diffeomorphism on $\mathbb{R}$, in which case the "length" just becomes $f(\lambda_2) - f(\lambda_1)$, which can be any number. Unlike spacelike curves and timelike curves, there is no notion of a parametrization by proper time/proper length, since the null equivalent is always zero everywhere on the curve. 
There is a commonly used length called the b-length or generalized affine parameter length which is used where, given some orthonormal frame $E_p$ at the point $p = \gamma(0)$ and a curve $\gamma : [0,a) \to M$, we have
$$l_E(\gamma) = \int_0^a \sqrt{\sum_i \left(g\left(\gamma', E_i\left(\lambda\right)\right)\right)^2}d\lambda$$
where $E_i(\lambda)$ is the $i$-th component of the basis which has been parallel propagated to $\gamma(\lambda)$, so that
$$\nabla_{\gamma'} E(\lambda) = 0$$
Consider for instance (2D) Minkowski space, with a "canonical" frame
$$E^\mu_i = \delta^\mu_i$$
and a null curve $\gamma' = (\lambda, \lambda)$, $\gamma' = (1,1)$. The g.a.p. length is
$$l_E(\gamma) = a \sqrt{2}$$
This kind of length for causal curves is used for instance in the study of singular spacetimes or to have a well-defined van Vleck determinant for the study of Green functions.
A: Spacetime manifolds are observer's manifolds, and there is no spacetime manifold all observers agree upon. However, according to the second postulate of special relativity, lightlike phenomena are observed by all observers as moving at c, so there is always observed a traveled distance > zero.
In contrast, the spacetime interval cannot be observed by observers, it may only be calculated, in the same way as the proper time of a particle. For this calculation, due to the structure of spacetime, all observers agree that the corresponding result is zero.
Example: In the following diagram where c is set to 1, a straight worldline of an object is observed which is moving 8 light minutes within 8 minutes - a lightlike movement.
As you can see, the observer observes a 45° worldline which is not zero. The zero cannot be found anywhere in the diagram, it is not part of the observation. However, he can calculate the spacetime interval, and he will find zero.

