Nezuko and Muzan

If Muzan were to consume Nezuko when she was in this half human half demon state would he still have been able to conquer the Sun, or would her being half human interfere with that since she can go in the sun either way.

Also, unrelated question, but when Tengen and Shinjuro were ordered to guard Nezuko and Kagayas son, did anything actually happen during that? I mean did any demon every actually find them and they had to attack it? Or did nothing basically happen?