3 Which child?
a It was not and is not a requirement to become a US Citizen And in most cases males became Naturalized US Citizens, which made the wife and any foreign born children US Citizens
a In German ü, with the 2 dots over it sound like English UE, so I would suggest you don't overlook that spelling... after all the family member who told you it was MUELLER in Germany was not living when they came to this country :) In fact if you search for MUELLER in immigration records and then MULLER you will find many more MULLER.http://en.wikipedia.org/wiki/M%C3%BCller_(surname
I find this WW I Draft Registration for
Name: Theodore Miller
Birth Date: 6 Feb 1876
Naturalized US Citizen (no birthplace listed)
Wife Marie he was a Farmer and lived at Elwood IA
Is this your Theodore? Note even if a man was an alien he still needed to register... and even if he registered dose NOT mean he served.