Comments:

MEL-LHR will count as one segment as long as it has one flight number.

You can add two segments in Australia or Europe. You are at 18 segments, and are allowed 20. However, any OneWorld routing from SIN-NRT appears to go via HKG and thus will be 2 segments, only leaving you one additional.

Do you really want to fly AA from JFK to NRT when you could fly CX from JFK to HKG? CX has the superior product. You might as well get as much pleasure as possible.

I looked up the latest pricing that UserMark has on BKK. It appears to me that the price has popped since FewMiles posted on his site. The price for AONE4 is 311590 baht, or $7615. Taxes and airport fees will add about $400 to $500 to that.