What must they do when practice approaches are flown here in VFR conditions, or during check rides?
The FAA won't enforce a local rule not published in an FAA document. The power the airport has over transient aircraft in such a situation is rather flimsy at best.
They they cannot make any such rules about what happens in the air unless it directly affects the ground (e.g., noise limits). Federal Supremacy, and all that -- see Allegheny Airlines v. Village of Cedarhurst, American Airlines v. Town of Hempstead, and Santa Monica Airport Association v. Santa Monica. An ordinance flatly prohibiting training or aerobatic flights which are in compliance with the FAR's in federally-regulated airspace is unenforceable. Of course, 91.307 prohibits aerobatic flight "Over any congested area of a city, town, or settlement", so that one is already an enforceable FAR.
It will be rather hard for them to identify the violator and bring that person to court.
In that case, the ban on training operations is essentially unenforceable since they would be discriminating against non-based training operators while not enforcing against those based there.
I won't disagree, but as I said, I don't think the FAA will take action against a pilot who violates a non-Federal rule. And I don't think the state will take action on their own if it's not in a state publication (which apparently this one is not). As for the city taking action, it's not going to be easy for them to identify the pilot and bring him/her to court and carry the necessary burden of proof. I'd be interested to hear how many times that city has done that (if ever). But I would not engage the person on the UNICOM, either.
No, I'm not. The FAA isn't going to enforce a local ordinance without an FDC NOTAM saying the same thing, and no other sort of NOTAM is regulatory from an FAA perspective. Anything else is merely advisory in nature, and there must be some underlying FAR on which to base an enforcement action (e.g., Runway NOTAM for runway closure due to personnel and equipment working, pilot lands on runway, pilot is done for 91.119 and 91.13, not for violating the NOTAM itself). The FAA would have a difficult (probably impossible) time showing that safety was compromised in a way that violated any FAR just because a pilot back-taxied on a runway -- in fact, 91.113(g) specifically exempts the case of a pilot attempting to clear the runway in order to make way for the next aircraft from the general rule that the landing aircraft owns the runway. If back-taxiing is the most expeditious means to clear, that is compliant with the FAR's.
Lots of them? Perhaps you can point out even one example of such an FDC NOTAM, as I've never seen one in all my years of flying other than temporary restrictions during runway construction or the like.
...and the FAA has gone so far as an emergency revocation on a pilot flying the pattern who deliberately cut off airplanes on straight-in approaches in violation of that regulation. While the question of courtesy and politeness regarding this issue may be the subject of sometimes spirited debate among pilots, the law and the FAA's and NTSB's position on this issue are not. See Administrator v. Fekete.
Advisory Circulars do not trump regulations, and there is case law on point. Don't bet your ticket on this AC on this point or you may end up like Mr. Fekete. Nothing wrong with asking the pilot on the straight-in if you can turn in front, but if you turn in front of them forcing them to alter their approach and they make an issue of it, you are the one on whom the FAA hammer falls, regardless of what that AC says.