Well technically there is nothing stopping you from doing so, but it isn't advised because it protects them from situations involving making hiring decisions based on race, gender, religion etc.
They will know eventually (sometimes online applications ask) because companies have to look diverse, they have to hire a certain amount of people from each race so it doesn't look like discrimination.
A resume is more like education and work history anyway, not the personal stuff. Thats what the interview is for.
Discrimination purposes I believe
Yup, this is right.