Implementing the example of Walter et al. (2007) paper
Initially and still I'm not so sure how to get the color of the object behind the glass into the formulas.


Normal, this was written with path / ray tracers in mind where that part is easy. Your idea of a cube map might be ok for some use cases (eg. reasonably rough surface for the distance to the surrounding objects). It is not a general solution however. That would likely require GPU ray tracing.


  Going by my first thought again I just multiplied the result of the BTFD with the sample of the o_t vector but apparently the BTDF was really big so the whole slate turned white (possible bug?).


I'm writing this from memory but I think you need to divide by the PDF. Which means that your weights will be close to 1. But not quite 1 as the sampling distributions don't include the shadowing terms.


  Also I think I read somewhere that the reflection and transmission should be handled seperatly depending on the result of the Fresnel formula. Or should it be handled at the same time (possibly somethink like : BRDF * sample_of_reflection + BTDF * sample_of_refraction)?


A classic path tracer would randomly pick either reflection or refraction based on the fresnel coefficient. That does efficient sampling. But if you can easily get both from a cube map, you can also blend them with the fresnel coefficient.


  After some playing around with this function I noticed that is basically something akin to a gaussian distribution pointing towards the positive z-axis. But shouldn't the distribution give samples which are distributed around the surface normal?


Yes, you need to transform the generated direction. It is mentionned in section 5.2, after equation (24), that $\theta_m$ is the angle between $m$ (microfacet normal) and $n$ (surface normal). They don't go into great detail as this way of defining microfacet models goes back to much older articles.

The short version is that you build an orthonormal basis using the surface normal as z and two other perpendicular vectors as x and y. Here it does not matter which as the distribution is isotropic, as long as they are perpendicular to each other and to the normal. Then you transform using that basis.