O=Cl=O → O^--Cl-O^- → (cyclic)O-Cl-O

This is an intramolecular reaction which isn't all that common.

The double bonds break, causing the O's to each have a negative charge. This forms an ion. The ion bends and eventually the 2 O's bond with each other forming a ring.

But would this really happen? I mean yes, sunlight is enough to split Cl₂ into 2 Cl radicals but would it be enough to break the O-Cl double bond so that the ion can form?

 

I have heard that this won't happen due to the negative charges. But really, this conserves octets and hypervalency(ability to have more electrons than an octet).

 

But surely this intramolecular reaction where the oxygens bond with each other is how cyclic peroxides form no matter the R group if you already have 2 oxygens bonded to the R group in the same place.

 

I mean, how else could you form a cyclic peroxide besides adding hydrogen peroxide to a molecule that doesn't have a peroxide group?

I don't have time to look through the bulk of your post, but you might be interested in dimethyldioxirane (DMDO). It's made from acetone, NaHCO3 and Oxone in water. The mechanism isn't too complicated.