The stairs are closed, each floor separated by doors. When standing under the stairs, it sounds as if someone left the door open....[Kompriband] is a good idea.
Impact sound is not a problem, the stairs are extremely quiet to walk on.
Not yet. Your question Which material for joint sealing between stairs and wall for sound insulation? should better be ~ for sound coupling . This is not a bug, but a feature. You can try compressible tape, that shouldn't hurt. But I share the doubts that the sound specifically looks for its way through there.