As noted above, those three hours are not "instrument training," but rather "flight training in a single-engine airplane on the control and maneuvering of an airplane solely by reference to instruments" -- a distinction which allows a CFI-ASE with no instrument instructor rating to give those three hours, i.e., to give all the training for PP-Airplane.
Correct.
All candidates for CFI-Airplane must hold an Instrument-Airplane rating on their pilot certificate. However, that does not give them instrument instructor privileges. So, you're talking about someone with a CP-ASEL-IA and a CFI-ASE (but no IA on the CFI), right?
That instructor (CFI) test will be for a particular instructor rating, just as the PP test was for the ASEL rating. I'll assume you really meant "Instructor-Airplane Single Engine (CFI-ASE)". At that point, the CFI can give training for PP-ASEL, CP-ASEL (except the required 10 hours of instrument flight training), flight reviews in ASEL aircraft, and complex training in ASEL aircraft. S/he may also be allowed to do HP/TW training, depending on his/her endorsements for those.
That would be the CFI-IA additional rating ride. Now s/he holds CFI-ASE-IA At that point, s/he can also give instrument flight training for the IR, IPC's, etc, in ASEL aircraft (no seaplanes, since s/he does not have an ASES rating on his/her pilot certificate).