Who usally handles bills on a senate floor

the majority leader

This person is the head of the majority party in a legislative body, especially the U.S Senate or House of Representatives, and:

takes care of the bills on the Senate floor.

the doctrine of nullification recommended that states living inside the union have the one-sided, inalienable ideal to void any law made by the national government that could be esteemed illegal.

the united states was shaped based on a general agreement among its individual states. hence, it is inferred that under the hypothesis of invalidation, since the states are the establishment of the union, they have the ability to referee and disprove unlawful laws.