You are correct, but modern receivers will actually use the transmitter like a "NDB" and will detect that the incoming position is not the same as the stated in those messages, ignoring them.

The same thing happens with sudden changes in GPS messages that are completely out of alignment with the inertial system, they will be ignored. (unless you somehow are capable of knowing exactly where the plane is and gradually hack the GPS messages)

ILS systems are maybe the most dangerous and prone to hacking, but keep in mind that those frequencies are under surveillance and police is dispatched within seconds of a clandestine transmission on those frequencies. Again, here the gps and inertial is also used to guarantee nothing completely wrong is accepted as valid ILS sources.

How are ILS frequencies policed? Does the airport have receivers in multiple locations which automatically detect a failure, and notify someone in the tower?

Exactly. Because the ILS range is short, it becomes easy to pin point location. Those protections are embedded in the ILS system itself

Collision avoidance is not a function of the transponder. ADS-B merely communicates the position of local traffic.

ACAS is responsible for determining the vectors for the aircraft and the local targets. If there is a collision path calculated, ACAS then communicates with the target's ACAS to determine which way each aircraft is going to move. The pilot is then responsible for executing this action.