Can you explain why it is considered a point-particle (unlike the proton), that is: having no spatial extention? Doesn't mass have to occupy space anymore?
The electron is an elementary particle, and thus a quantum mechancial entity, of the standard model of particle physics.
The proton is composed out of elementary particles shown in the link in a table.
Quantum mechanical entities are described by solutions of quantum mechanical equations, Dirac for the case of the electron, plus the field theoretical solutions of the lagrangian of the standard model linked above. These solutions give the probability of finding the electron at (x,y,z,t). Depending on the boundary conditions of the problem at hand, this probability might be constrained on a point, if the dimensions are macroscopic, or display wave like characteristics.
In a recent answer user AnnaV showed that it has even been filmed, why do they think it must be definitively smaller than 1/10^-16 cm? How can such a lot of mass be squeezed into a vanishing point?
It is the way the mathematics of the standard model works out. The particles in the table are entered as point particles in the lagrangian, the mathematics is cranked, and predictions for the behavior of the electrons are made. These predictions work to a great accuracy, as demonstrated by the four LEP experiments..
That is the why, because the predictions of having the electron a point particle fit the data.
Now theory advances and predicts that they are not point particles but have a one dimensional extent, at values smaller than 10^_33cm they are strings. With the new energy of the LHC it may be possible to start seeing the string effects. It may also be necessary to wait for higher energy machines to see whether elementary particles are really strings in many extra dimensions. At the moment the definition rests on the standard model of particle physics and its successes up to now.
That a particle decays into other particles is completely disjoint from it having substructure/being fundamental or composite.
Some examples: A highly energetic photon may "decay" into an electron and a positron in the presence of another object that takes the excess momentum. That doesn't mean a photon is a composite of electron and positron. A free neutron decays into a proton, an electron and an electron anti-neutrino with an average lifetime of 10 minutes, yet it is a composite state of three quarks.
Being constituted of other particles means being a bound state of these particles. Quantum field theoretic processes have no problem turning one kind of particles into other kinds of particles (subject to certain rules, of course), but this sort of process does not imply that the results actually constituted the input. In no meaningful way is a photon a bound state of electron and positron, in no meaningful way is a neutron a bound state of proton and electron, and in no meaningful way is a muon a bound state of an electron and neutrinos.
Best Answer
Believe you me, people have devoted a lot of time to coming up with composite models of the electron, without much to show for it. For example, see the preon.
High energy scattering experiments have shown that the charge radius of the electron is very small, and yet the rest mass of the electron is also very small. It's difficult (though not impossible) to achieve both in a composite model.