I don't think efreets are counted as elementals.
About the boss monsters, I think you have to kill them with direct damage to get the count (the killing hit), provo wont work, but I don't know if you get counts if using tames. But maybe there is some other issue with those!
About the boss monsters, I think you have to kill them with direct damage to get the count (the killing hit), provo wont work, but I don't know if you get counts if using tames. But maybe there is some other issue with those!