Is there a rule of thumb for knowing how organic peroxides will form? I thought the peroxides bonds appeared where an oxygen is double bonded (AP, MEKP), but in benzoyl peroxide, it leaves the double bonded oxtgen unaltered. Also, what makes a compound able to become a peroxide?

Peroxide implies that there are two oxygen atoms have both bonds with each other and both also have one additional heterogeneous bond.
Any molecule with such a chain is a peroxide.

Peroxidation happens when a stable and electrophile atom/chain/compound is exposed to a mono-hydrogen peroxide anion, the electrophile atom will connect with the peroxide chain.

Nucleophile nature can be induced by protonating an oxygen (C=O -> C-OH+ -> +C-OH) atom (as in AP synthesis), but doesn't have to happen like that.

Thanks, Mokothar, but your reply only made me realize how little I actually know about chemistry. Among other things, I'm confused about the protonation of the C=O. Does the H+ come from the acid in the mixture or from a dissasociation of the H2O2 into H+ and HOO-?

Both of those happen, and the "keton" carbon won't distinguish, but there should be way more H+ present from the acid catalyst.

I'm not sure on how exactly HOOH +H2O goes to HOO- and H3O+ though

I think I get it. Thanks Mokothar.