There are such things as “radical anions”, which bear both a radical and a charge. One example is the ketyl radical anion that is formed from the reduction of benzophenone by sodium metal. https://en.wikipedia.org/wiki/Ketyl

It’s simplified. The interaction is between a lone pair on oxygen and a half-filled orbital on carbon. There are only 3 electrons in the system. The middle section depicts the lowering of the energy of the lone pair, and the raising of the energy of the antibonding orbital. Overall the system is more stable.

For our purposes, no.
In fact the cyclopropylmethyl radical is unusually stable, and although I don’t have the numbers at the moment the cyclobutylmethyl radical would also be more stable than an ordinary primary radical as well since it can interact with the “banana bonds” of the cyclobutane.

Yes PSf certainly would form a radical more easily since it has a chromophore that can be accessed with UV light. You’d get loss of SO2 and formation of two adjacent radicals which could recombine or go on to other chain reactions, depending on conditions.

I wouldn’t call it hyperconjugation since hyperconjugation is (by definition) between sigma bond and and empty or half empty orbital. A quantum phyical chemist would call it just conjugation. Even though it’s very unorthodox.
[Modern Physical Organic Chemistry By Eric Anslyn]