There isn't really the same concept of "seniority" in general US culture as there is in other cultures. (I say "in general" because there are different varieties of US culture. Southern culture is more formal than Northern. Different ethnic groups are going to have different norms. Groups like the mafia probably enforce status differentials more than the general population does.) There are very few cases where using any name would be considered rude. Maybe if you're in court, the judge might consider it rude to refer to them by their name rather than as "your honor".
There are cases where it's considered proper to refer to people by their last names. And referring to someone by just their last name generally comes across as casual; in formal situations one uses a title or honorific before the name.
There are two general categories of where last names are expected: a formal situation where everyone refers to each other by their last names, and a situation where there is a large hierarchical distance where the "higher" person is addressed by their last name.
Although the Harry Potter books takes place in a British boarding school, so it's more formal than the general US social situation, it's an example of different norms that you might see in place. You'll notice that all the students address their professors as Professor Lastname, and the professors address the students by just their last name, while in conversations within just students or just professors, you'll see first names. So between professors and students, the interactions are formal, with the difference in social level being marked by the students using the title "Professor", and professors just using last names.
In the US, prior to college there is a clear gulf between he social status of teachers versus students, and teachers generally refer to students by their first names while students refer to teachers by honorific + last name. The honorific will usually be Mr./Mrs./Miss/Ms., but if someone is teaching high school despite having a PhD, they might have the students refer to them as "Doctor". In the other directions, in less formal/"progressive" schools, the students may refer to the teachers by their first name.
During college, there is less of a gulf, especially in graduate schools. Whether a doctoral students addresses their adviser first name will depend on their relationship, although they will probably refer to them by their last name, even if they are addressing them with their first name. A TA will probably address the professor they are working for by their last name, but if the professor is less formal, the TA may address them by their first name. Students will generally address, and a TA who asks their students to refer to them by their last name will probably be seen as pretentious.
However, you have more leeway for asking people to refer to you by your last name if you also refer to other people by their last name; then it's an issue of formality rather than relative status. An associate professor is indeed a more prestigious position than assistant professor, but it's not enough of a status difference to justify last name status. Unless they refer to assistant professors by their last name, an associate professor who asks assistant professors to use their last name is going to be considered a pompous jerk. In US culture, marking someone as being a lower status is a faux pas more often than not marking someone as being a higher status is.
The military is one area where there is a rigid hierarchy. From what I gather, people will generally be referred to by their rank and last name, but in some cases they'll use just last name, and in casual settings they'll refer to each by their first name.
Also, never refer to a woman by her first name if you would not, in similar circumstances, refer to a man by his first name.