So you wouldn't have any issue with the measles vaccine if it was in a separate shot?

They don't have the autism issue in Japan. They still use Measles vaccine.

Does that apply to the Mumps and Rubella shot?

Mumps is not deadly , can cause sterility in adult men who develop it. My nephew was conceived and born after my brother was quarantined for the mumps in college ( his 3rd child).

Rubella also has low fatality rate but can cause deafness in child if the mother develops during pregnancy. Shouldn't we be vaccinating young women before child bearing age ?

Both vaccines exist as separate also. More expensive.