1) Yes - no problem, the two stopovers can be whenever you like. Obviously at least one of your trips through London must, therefore, be a transit, and two of them if you want to stop in Muscat.
2) It counts as touching the continent. Some years ago one or two people reportedly succeeded in getting that sort of thing ticketed, but not recently.
3) I assume you meant LAX-SYD-MEL-PER-AYQ-SYD. In which case you have one unused segment (of 20 allowed). The whole thing isn't very mileage-optimised if that's important to you - you could rearrange the cities in a different order for more miles.
4) Can't help on that one - I haven't ticketed ex-ARN.