Welcome to the slightly esoteric RTW world.

Your itinerary looks perfectly legal to me, and would count as an "AONE3" as it's in First and would involve 3 'continents', as defined by OneWorld: N. America, Europe/Middle East, and Asia. You can finish in LAX if you wish, or add up to 6 other N.American flights in First for the same price. These flights can be open-dated, and can be taken at any time (assuming OK availability) up to 12 months after you departed DFW in the first place. (The rules for these extra flights are a little more complex, but the major limitation is that you can only have two stops more than 24 hours, and I guess one of them would already be taken up in LAX.)

Yes, the rule is that the trip cannot be completed in less than 10 days, beginning to end. Your three week "land segment" will easily take care of that.

As for availability, it should be noted that these A-class tickets are not quite as flexible as full-price F. If you want to change at the last minute, there could occasionally be an availability problem. That said, there are A-class seats available on 2 of the 3 AA flights DFW-LHR for the next few days; full availability on HKG-LAX, and A is the same as F on CX from DXB to HKG. (Some DXB-HKG flights are already full in both F & J.) Your biggest potential availability problem doesn't appear to apply in your case. That is the D class on BA from LHR to IST. (When there is no F, you go down to D, not J.) D class on BA is often full by the day of travel, but so long as you don't want to change right at the last minute, you'll be OK provided that you reserve a reasonable time in advance.

Edited to add: If you have an OK A-class reservation, you have just as much right to a seat in the First Class cabin as any F-class ticketed passenger. I personally am not aware of downgrading of A-ticketed pax having happened, although I guess it's possible in extreme circumstances, e.g. the President's wife demanding a seat. Has this happened to any FTers?