@ElenaParent: see some career destinations after graduation for each major:
https://career.berkeley.edu/Survey/2017Majors
For those who are going to grad school are:
Applied Math: ~20~30%
Math: ~40~50%
Taking honors classes are certainly not required for graduation with honors. In fact, graduating with honors is technically irrelevant for grad school application purpose since one would typically be applying for grad schools before you officially graduate with honors.
I’ve done a honors class for physics, and the succinct experience I can share is it’s for folks who would like to take on the extra challenge, strictly optional. I do not see playing in admission to PhD programs.