As regards AA codeshares in Asia, you can only book the codeshare in connection with a flight to/from the U.S. This means that you could fly, say, JFK-HKG-SIN all on AA codeshares operated by CX. After this, you won't be able to avoid CX or JL as marketing carrier. Anyway you should get 70% of miles on JL IIRC. So fly AA codeshare to NRT or HKG and a further destination beyond the gateway and then use JL in Asia.

The same could work for your last inter-Asia segment connecting to your AY flight to HEL. However I am not aware of whether AY has any codeshares to/from HKG or NRT. If you find some, you may route xxx-HKG/NRT-HEL all with AY as marketing carrier.

The ideal route would therefore be sthg like ...xxx AA HKG AA xxx JL NRT JL xxx AY NRT AY HEL...

In Europe AFAIK LA codeshares are bookable only in connection with a flight to/from Chile. Therefore you cannot book MAD-LHR on LA as a standalone segment. You have to book with IB.

Edited to add: I played around with the oneworld interactive map and I could find one single AY codeshare on the HKG-BKK route. Unfortunately, the same does not exist in the opposite direction. So this would be an option for an eastward RTW (HEL AY HKG AY BKK), but not for an westward one. Flight in question is CX713/AY5843.