Is there a difference between a (under)graduate and honours studies - also concerning thesis?
Just about every possibility exists somewhere in the United States. There is no one answer. 

Some colleges or universities do not require any type of thesis for an undergraduate degree. Others may require students in an honors program only to complete a thesis or project. Yet others may require all students to complete a thesis or project for an undergraduate degree. 

At the graduate level, there are research master's degrees and non-thesis/course-only master's degrees. The latter may still have a project or paper. A research master's degree is usually for those who are going on to a Ph.D.