GFCI connection and trip
For downstream outlets from a GFCI to work correctly, both the hot and the neutral on the circuit have to pass through the GFCI. A GFCI works (and this is a simplified explanation) by detecting the difference in current passing through the hot and neutral.  If it has current moving through the hot and nothing through the return, then it trips because it detects that as being an alternate ground path. I highly suspect that the downstream outlets are either connected to a separate neutral or ground somewhere.