A few comments:

You can travel on OWE MEX-DFW (or MIA)-LIM-SCL rather than buying a separate ticket.

If you can't book SCL-AKL when you want to travel consider changing to SCL-SYD-AKL-SYD-BKK.

The way you have laid out the itinerary it appears that you are only stopping in CAI and FCO, however, you do have five segments in Europe and are only allowed four. Looks like the only way to fix this is to drop either HEL or FCO from the OWE itinerary (i.e. ticket separately).

You will run into problems due to the 20 segment limit if you choose to fly MEX-DFW-LIM-SCL but you will be OK if you drop HEL or FCO. Also, you can cut the number of total segments by going SYD-NRT-HKG-BKK.