When used with a する-verb (i.e. meaning "to do something"), is it correct to say that: ...
する is always transitive, because in this case, the する-verb is the "object"?
E.g. in 運動します, "I" am the acting agent and I "act" (します) on the "object" (運動)
-- OR --
The transitivity depends on the use of the する-verb?
E.g. 運動します is intransitive while ピアノを練習します is transitive
If you are asking about the transitivity of a resultant combined verb, it depends on the combination. For example, 運動する and 死亡する are always intransitive, while 保管する and 破壊する are always transitive. Some verbs such as 移動する and 実現する are ambitransitive (i.e., they can function both intransitively and transitively). You have to remember the usage of each combination.
If you are asking about the transitivity of する itself before it attaches words like 死亡 and 愛, I'd say you shouldn't attempt such a classification. We never say 死亡をする or 愛をする in the first place. Let's not think of this type of する as a verb on its own. Instead, think of it as a special suffix that can turn another word into a verb.
