Spayed means a female dog has had her reproductive organs (ovaries and uterus) removed. Neutered means technically that an animal has been de-sexed, but usually is used to refer to a male dog being castrated.
A complement is a word or word group. that completes the meaning of a. verb. Every sentence has a subject. and a verb. Sometimes the subject. and the verb can express a. complete thought all by themselves.
Catenative complements are non-finite clauses, not direct objects (which are usually noun phrases). Simple catenative constructions have just a non-finite clause as complement. The issue of whether forms ending in -ing (like smoking) are verbs or nouns is a bit tricky. He stopped smoking is strictly speaking ambiguous, but the verb is the more salient interpretation (He stopped smoking cigarettes).