The reference location for any ILS installation is the localizer antenna array. This could be several hundred metres from the DME and as such GPS...
Alphacentauri, not quite. (Yes it's not allowed, as i said.)
For any ILS, there are two reference points in the GNSS database. One is the LOC transmitter as you say. The other is the GS transmitter, which is (approximately) where the DME is. It is possible to select the correct reference point such that the GPS mimics the DME, although of course that would be illegal in Aust! (Not always elsewhere...)
Yes, if you picked the wrong waypoint it could be 2000m or more away from the DME. So? The sole purpose of a DME in an ILS installation is for an integrity check. If it's 2000m or more out, the integrity check will fail. The AIP (rightly) provides for this, but there is zero safety implication associated with an erroneous distance readout
unless the GS signal is faulty. Hence the need for this integrity check.