It seems obvious that these schools recruit kids specifically to play sports. Is it on the level? Are there any rules against it or the manner in which it is conducted?
Yes, there are rules against it, or at least rules that seek to limit it. That's why when a star athlete from a NJ public HS (like Rutherford HS) decides to transfer to a state parochial power (like Bosco), he'd be expected to sit out a few games before competing as the new QB (or WR). Of course, whether the student in question was "recruited" is a matter of perspective.
While any official representatives from the parochial power are not allowed to contact the player directly, there is no rule saying the player cannot contact the school. Moreover, other student athletes and parents who do not work for the school have nothing stopping them from encouraging public HS star athletes to transfer.
What is there to stop a team parent or any fan of a parochial power from (after seeing an impressive athlete perform in a Pop Warner game) approaching a kid's parents, praising his talents, and inviting the family out to dinner in order to inform them of the benefits of playing football at a school like BC, DBP, PC, SJR, SPP, etc....
It happens all the time. While the fans in these cases are essentially functioning as unpaid scouts, they are still not employed by these schools. Sure, they may be "on a mission" for coaches expecting contact with these players, but good luck proving it. Sadly, given the over-emphasis on football at some of these parochials, the cleverness with which they skirt the rules is on track to one day rival the cunning of the SEC. But hey, money talks; and the success on the gridiron enjoyed by these schools has resulted in increased donations. If that's the way things work, can we really blame them?