How can a point-particle have properties? I have trouble imagining how two point-particles can have different properties.
And how can finite mass, and finite information (ie spin, electric charge etc.) be stored in 0 volume?
Not only that, but it can also detect all fields without having any structure.
Maybe it can check curvature of spacetime to account for gravity, but how can a point contain the information of what the other fields-vectors are? This seems to mean that also the information/volume in space is infinite.
Mathematically, a point cant have any intrinsic structure, so how does physics which is a mathematical theory explain this?
 A: Particles are never detected as points. The more precisely you want to measure the coordinates the more energy you would need. This means that to determine a particle's coordinate exactly you'd need infinite energy. In all other cases the particle appears as a wave package stretched in space, you only know the volume area where it is located.
When someone says the particle is point-like, he means that the dependence of the measurement precision on energy is smooth and logarithmic, and potentially one can localize the particle to any given however small volume given enough energy.
A: The assumption of particles being points is a modelling assumption used extensively in mathematical physics. 
When the orbits of the planets are calculated, the center of mass points are used in the first order calculations and assigned the mass of the whole planet. The sun too is a point at first order. Nobody has trouble with that.
Similarly when solving Schrodinger's equation for the hydrogen atom , as a potential well for the electron.
In second quantization the "particle" has a structure of Feynman diagram loops of all orders accompanying its probable position as particle. In strings even the Feynman vertex is no longer a point but has extension. 
A: When one says that an elementary particle is point-like, one is referring to the fact that theoretically, there's no limit to how small a region a detector can localize a particle to. For the sake of argument, let's imagine two wrong things (a) that such an ideal detector is possible and (b) complications arising from Planck scale physics don't change anything conceptually.
Even if you allow for that, your worry that information is being stored in a zero volume region is still unfounded. It would be a legitimate worry in pre-relativistic-QFT physics. But we know particles aren't pellets that move around carrying information. They can disappear and be spontaneously created out of the vacuum. What this is hinting at (though some might prefer a different picture) is that particles aren't fundamental - fields are.
The quantum fields for various particles are defined everywhere in space. Once you specify what kind of structure the quantum field is (a scalar, vector, spinor, etc.) and what its other properties are (say, the symmetry group under which it has local gauge invariance), you have specified what spin, charge, mass etc. its particle excitations will carry. Since the field is defined everywhere in space, there's plenty of room for all that information. So in a certain sense, the information that a detector detects is encoded everywhere in space (because the field is everywhere) - and the specific structure of the detector just picks out the right information you asked for.
Finally, two point-particles (say an electron and a muon) have different properties because they are excitations of two different fields defined everywhere in space - and the detector you build specifically for the electron will pick out the "signal" from the electron.
A: My feeling is that, at the foundations, the information about a particle needs to be encoded in its propagator. That is, all the coupling constants, mass, spin, and other information needs to show up in the propagator.
That means that instead of the information being contained in a point, instead the information is contained essentially in how the particle moves from point to point.
Another way of saying this is to note that what we call a "particle" is an effect seen by our experiments, where nature does something that seems to be repeated in a consistent manner. Relativity requires that an object be able to move, hence the propagator.
A: Susskind says (in his Stanford lectures on string theory) that subatomic particles are not point particles, they have  a spatial extent. This does not depend upon his contention that they are really extended string objects. It's just a consequence of the fact that even the electron is "fuzzy" due to being embedded in a little cloud of virtual photons and virtual electron-positron pairs. And compared to the electron, the proton is "huge".
