BA award search covers all oneworld carriers, including CX and JL.
Yes, but not in the calendar view, which is what your reply to
Ducati highlighted.
QF probably has the most useful calendar view at the moment.
between AA, BA, QF (and JL for CX sometimes) it is relatively easy to search for oneworld award space yourself for free.
Absolutely. BA is where I usually start, unless I know that AA and BA are my only options (in which case AA.com is easiest).