Why do we use spinors for describing fermions? I.e., what properties of the spinors gives us a reason for using them for describing of wavefunctions of fermions?
 A: We're using spinor fields for fermions because Nature does the same. Nature does so because She has no choice. Pauli has proven the spin-statistics theorem that says that all fields whose particles obey the Fermi-Dirac statistics (with the Pauli exclusion principle) have to carry a half-integral spin; and those with the Bose-Einstein statistics have to have an integral spin. Mixing a spin with a wrong statistics or vice versa would lead to negative prrobabilities or energies unbounded from below.
The only half-integer field that doesn't require any gauge symmetry to get rid of the negative-norm states is the spin-1/2 field, the spinor. A spinor may be viewed as an object that is more elementary than a vector and that was previously overlooked. It's also possible to build vectors and tensors out of the spinorial components – vectors and tensors may be represented by spintensors of various sorts. But the word "spinors" should be reserved for the representations with $j=1/2$.
There also exist "unphysical" field theories such as the topological ones that may violate the spin-statistics relations. Also, Faddeev-Popov ghosts used to deal with gauge symmetries in a modern way always violate the spin-statistics relationship – the rule is exactly reverted for them. $b,c$ are fermions with an integer spin for a bosonic symmetry, and $\beta,\gamma$ are bosons with a half-integral spin. They don't create physical particles.
