1. Normal. Pushing on the horn button grounds the circuit causing the horn to sound.

2 & 3. Bad instrument cluster ground. The cluster is ground via the center screw on the plastic bezel face surrounding the instrument pods. That ground is notorious because a loose screw, or rust behind there can made the ground ineffective. Also, the thin ground strap to that screw location from the back of the instrument cluster can be bent or broken off easily by a previous owner. Many of us have run a second wire from a ground at the back of the instrument cluster as a backup.

4. Don't know. There is a separate relay for both systems. But the bad ground could also be affecting them.