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2

No, “this conclusion” is based on the topological properties of rotation groups. Namely, for any $n > 2$ $\mathrm{Spin}(n)$ is the universal cover of $\mathrm{SO}(n)$, whereas for $n = 2$ it is not. That’s why in $n > 2$ any thing has to be controlled by a representation of the Spin group, whereas in $n = 2$ it has not.

1

The answer is 'no' in condensed matter also. As anyons are neither bosons nor fermions, they can follow some statistics other than BE or FD, but it has nothing to do with fractional charge. just-learning has already given you a perfect example.

5

Not sure about the condensed matter context, but in general the answer is NO. For instance, quarks have fractional charge but are regular fermions.

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