Yes, it makes sense to talk about the transitivity of verbal noun plus する constructions. I would simply link you to the introduction of The Light Verb Construction in Japanese: The Role of the Verbal Noun, but I suppose it's better if I repeat some of the examples here. All of the following examples are taken from page 8:
First, intransitive verbal noun + する constructions:
- 到着する arrive
- 誕生する be born
- 死亡する die
Second, transitive verbal noun + する constructions:
- 警告する warn
- 破壊する destroy
- 攻撃する attack
Third, verbal noun + する constructions which can be either transitive or intransitive:
- 移動する move
- 破損する damage / be damaged
- 紛失する lose
- 縮小する shrink / become shrunken
- 終了する finish
- 増加する increase
No, を arguments don't need to be final; they can be scrambled out of their basic position. Because most constituents in a Japanese sentence are marked (e.g. with case markers like を or postpositions of some kind), sentence order is relatively free, although certain orders may differ in emphasis, and the order isn't entirely free.
If you want details about sentence order, I suggest you ask a separate question about it.