I never experienced any of what you say as a child. We travelled internationally a number of times, never had anyone tell us she wasn't my mom or anything.
I am not sure what you mean about the money thing. My mom was on my accounts when I was a kid (with different last names) so she could send me money. As an adult, I can't see how sending money would be an issue. My mom and I transfer money to each other fairly often still ($70k recently, went through fine).
I did not stay in my home town. Not sure how they would be relevant.
Again, most people would assume (if they assumed anything) that my parents were divorced, which is incredibly common. Half my friends had different last names from their siblings and/or parent. Blended families are incredibly common.
I am now a dad of two. They have my last name, while my wife (their mom) has a different name. Again, never a problem at doctors or school or anything. They always make you fill out your full name and relationship. Again, super common to have different last names here im California.