Why non integer spins obey Fermi statistics?
Why is fractional statistics and non-Abelian common for fractional charges?
Both fractional/non-Abelian statistics and fractional charges come from the same origin: long-range entanglements. This is why fractional/non-Abelian statistics common for fractional charges.
One way to realize long-range entanglements is through the string-net liquid phase of a pure bosonic model. The ends of strings in string-net liquid are non-local and are topological defects. They can have fractional statistics. The ends of strings can also carry fractional charges/spins due to the same reason: the ends of strings are non-local and are topological defects.
Half-integer spin particles obey Fermi-Dirac statistics and integer spin fields obey the Bose-Einstein statistics – it's true because of Pauli's spin-statistics theorem.
Concerning the second question, I suppose you meant fractional spin, not fractional charges. In the case of 2 spatial dimensions, the trajectory of one particle around another is non-contractible (if we allow to cross the other particle) so even 2 rotations fail to return us to the original state. That's why the wave function may change by more than by the sign: it may pick a general phase (fractional statistics) or it may even be multiplied by some general unitary matrix (non-Abelian statistics). See Fractional statistics on Wikipedia.