Is it permissible to touch the inner wall of the Kaaba within the Hijr of Isma'il, and to press one's chest, arms, and cheek against its walls from inside the Hijr, out of reverence for it and not seeking blessings from it?

Rubbing and clinging to the walls of the Kaaba, other than the two Yemeni corners, is not part of the Sunnah. The Prophet, peace and blessings be upon him, only touched the two Yemeni corners and not the Levantine (Syrian) ones. He touched and kissed the Black Stone, and he touched the Yemeni corner but did not kiss it. He prayed at Maqam Ibrahim but neither touched nor kissed it. Furthermore, this rubbing is not an act of veneration for the House; rather, veneration lies in worshipping Allah within it, as prescribed by His Messenger, peace and blessings be upon him. This includes circumambulating it, touching the two Yemeni corners, and kissing the Black Stone. Mu'awiyah agreed with Ibn Abbas that acts of worship must follow what the Prophet, peace and blessings be upon him, prescribed, and no one has the right to legislate an act of worship based on their own opinion, even if they perceive it as an act of venerating the rituals. What many ignorant people do by rubbing against all the walls of the Kaaba and all its corners has no basis and is an innovation (bid'ah).