Baby born to two non-Taiwanese Americans - what paperwork?

My wife and I are both American citizens and neither of us has Taiwanese citizenship. We just had a baby and I’m not quite sure how to proceed with all of the paperwork. I fully understand how to apply for his American citizenship. However, when it comes to his legal residence in Taiwan (visa?, ARC?) and his National Health Insurance I really don’t know how to proceed. Any help would be much appreciated.

Also, one other question. My wife was born and raised in the US. She has always been an American citizen and has never held Taiwanese citizenship. Her parents obtained US citizenship more than 20 years ago. Last year they came back to Taiwan and renewed their Household Registration. My wife’s father says that my wife can now obtain Taiwanese citizenship through his Household Registration. I believe she can, but is it legal for someone who is already an American citizen to obtain another country’s citizenship without first renouncing American citizenship? And how does Taiwan’s government view this kind of situation? Thanks again in advance for any help!

I think the problem may be in Taiwan
You cannot assume ROC citizenship without renouncing your other nationality

but the question is
Was she already an ROC citizen at the time of her birth ( since her father was an ROC citizen), therefore she does not have to assume citizenship of the ROC since she already has it?

I would contact the MOFA or relevant department

If the parents have residency in Taiwan, the infant can (and must) get an ARC based on the existing family relation. You should go to the local foreign affairs police station and complete the paperwork as soon as you get the infant’s passport.

You will need photographs of course.

As regards the nationality question, it sounds like the wife should be able to get ROC nationality based on the relationship with the parents. Get all the paperwork together, including official copies of everything that seems relevant, and go to the local household registration bureau in your location of residence.

The USA recognizes the existence of dual nationality.