BA booking EDI-LHR-SIN-MEL with SIN-MEL on a BA codeshare QF flight.
AY may be another option. I would suggest you just play around with ITA Matrix and the "/alliance oneworld" option. Any option will be over 10,000 miles.
Yes but 10,000 butt in seat miles may not equal 10,000 elite qualifying points, which is what is needed for the challenge
Depends on the booking class
OP needs to look at the BA earning chart on aa.com to know this (or the AY earning chart if flying AY)