Hi Rich,
Thanks a lot for the information! It is interesting that now there is a CustomerAddress API.
Please note that I have managed to update the addresses successfully as I wrote in my previous post. But the problem I am having is a different one. It has to do with the old information being returned by the call to the Update method. Also the get customer method seems to also get the old information if it is executed several seconds after the update, as if the old information is being cached somewhere. Please read my previous two posts for more information.
Still waiting for Shopify to comment on this.