Few Questions

Not every piece of armor has the "heavy armor" flag checked. Notable exceptions are the Snow Muffler, all the animal suits(Chocobo, Nutkin, etc.), and I believe the Cat Hood as well. For example, Terra cannot equip any of these even with the Merit Award.

As for the Tintinabar, to my knowledge, it was hard coded to the item ID. Thus, regardless of what you do with the ME, the Tintinabar will have the "gain HP per step" effect, and nothing else will have it even if you check the flag for it, because the flag doesn't work properly. Lenophis's patch(I assume) fixes the flag to work properly and allow you to give anything the "gain HP per step" effect, as well as removing it from the Tintinabar if you desire.

The Esper thing is beyond my knowledge and probably requires more work.