#12 is no good either. The subject of the nonfinite clause (having...) must be the same as the main clause. This, on the other hand, is OK: [The girl] Having worked hard, the girl gained a prize.
Could it also work for other participle subordinate clauses like ones that start with '-ed'? I think there are three participles, present, past and another which I can't recall the name. I think all three can be used to start a subordinate clause. Right? For all three, if they are workable, the subject of a subordinate clause must be the same as a main clause?
Are you perhaps thinking of the perfect form as the "third form"? Having prepared so long and hard for the exam, Jane was quite disappointed when she received a C.