In Italian "piacere" originates from what in English is the object:
Subject: i gatti (cats)
Verb: non piacciono (aren't liked)
Indirect object: a lui (by him)
And that's how it comes to be; the sentence could also be written in the order I listed them, and also as "I gatti non gli piacciono".
Nope, 'gli' is 'a lui': the full list in the usual order from I to they is mi, ti, gli/le, ci, vi, gli.
The pronoun particle 'si' is special and it has three main uses:
Reflexive: it substitutes both forms of 'gli' when the object is the same as the subject, e.g. "lui si lava" (he washes himself), "loro si amano" (they love each other), "lei si è comprata un vestito" (she bought herself a dress)
Impersonal: it makes the subject generic, e.g. "si dice" (people say), "si mangia bene" (one eats well, i.e. the food is good), "si usa" (people use or are used to, i.e. it's custom).
Passive: similar to the previous, it makes the subject the object of the action, e.g. "si vendono biglietti" (tickets are on sale)
In this case, if you were to say "I gatti non si piacciono" it would mean "cats don't like each other"; to generalize you'd have to remove the pronoun entirely: "I gatti non piacciono" (cats aren't liked).