I live in high rise condo with identical units on each floor. Only when my neighbor directly above me flushes his toilet does my commode run briefly. My toilet functions correctly. Our building engineer has no answer for this.
It's normal. Your "out pipe" and his share a connection. As the water rushes past your connection, suction is created, and a little water goes "out" your toilet. I would get one of those "refresh" kits that replace the flapper, just to be sure and call it done.
Coteyr has the right idea, wrong pipe. The water supply pipe is constricted somewhere below both your apartments. Probably from scale buildup.
His toilet's demand is causing a dramatic pressure drop, which is unseating your toilet's supply valve. It's running a little bit, and you're hearing that. It may be causing water to run out the tank overflow and into the bowl.
It may be exacerbated by difference in the toilet valves on the two units.
The bathroom faucets may not be on the same pipe, so maybe no help there.