Is this even that important in college admissions?

Depends on the college.

Many Hispanic/Latino people have mainly white European ancestry (e.g. Spanish people), and many do not have obviously Spanish-derived names.

The only specific rule that I have heard of for Hispanic/Latino is the NHRP one “Are at least one-quarter Hispanic/Latino” (see https://collegereadiness.collegeboard.org/psat-nmsqt-psat-10/scholarships-and-recognition/national-hispanic-recognition-program ), although I have not heard of how they verify that, if they do (and have never heard of DNA testing for that purpose, especially since Hispanic/Latino people can be of any race or combination). Most colleges just take your word for it.

Again, usually the honor system. Ask yourself the questions in reply #2.

However, the Census definition of black or African American is “A person having origins in any of the Black racial groups of Africa.” Non-black North African people (e.g. Arab) are considered white. See https://www.census.gov/topics/population/race/about.html .