I'm going to try to make it as simple as possible.
Managed APP means you do not need HDG/VS or TRK/FPA inputs on the FCU (ILS-VOR-NDB-GPS-RNAV-RNP-RNP AR) The APP P/B must be on.
Manged/Selected means you only make FPA inputs( ILS-LOC-VOR-NDB)
Selected means you must do TRK/FPA inputs on the FCU (ILS-LOC-LOC B/C-VOR-NDB) and have raw data displayed on the ND for the PF
There is NO bearing on whether the A/P is ON or OFF except which Pilots does the inputs on the FCU (PF-PNF) and LOW VIS OPS (where the AP is required for AUTO LAND).
The Aircraft that have an OEB affecting the possibility of doing a NON PRECISION MNG APP can do a Selected APP with A/P on or a Managed APP with A/P off.
Airbus recommends doing all NON PRECISION APP's with TRK-FPA.(Anything except ILS)
There are minimum requirements to conduct a VOR, NDB, GPS, RNAV, RNP or RNP-AR MNG APP's, check your FCOM. Minimum must be GPS PRIMARY or ACCURACY CHECK positive.
And yes, a Selected ILS can be done with HDG/VS intercepting the LOC and GS using the FCU without arming the APP or LOC P/B's.
Guiones