Thank you for your comments and kind advices. They are all truly welcome and appreciated!

I should probably have mentioned that LAX is a transit of less than 24 hours, so I guess that makes the ticket in its current form valid, right?

I didn't know about the possibility of finishing the RTW ticket in another country than where we started. I'll definitely look into that. Thanks for pointing that out!

Now that you see the outline of the route, are there any obvious flights with better equipment I should consider?
-It doesn't really matter that much which countries we fly to between continents, as long as the visiting order of continents is unchanged. (Like I wrote earlier, we'll probably buy some separate tickets to travel between countries within the same continent).

If we can get extra miles on top of that, that's just icing on the cake.