There is no difference in availability, AA.com is just better at finding multi-segment itineraries. Any SAAver level award inventory can be booked with avios also.
Yes no problem.
Also, the cash element is often cheaper when using AA miles (compared to Avios). I recently booked 2xJ one way PER-LHR on CX for 120k AA + £90.
Booking the same flights via BA Exec is 200k Avios + £140.