What is the ruling on touching the Kaaba (or its door) and praying there while doing so? Is this Bidah (innovation)?

Praise be to Allah, the Lord of the Worlds; and blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
It is confirmed by sound Hadith that the Prophet used to kiss the Black Stone or touch it with his hands. He also used to touch the Yemani Corner with his hands. It is also a Sunnah to stand at Al Multazam (what is between the door of the Kaaba and the Black Stone). Abu Dawood reported in his Sunan that Abdullah Ibn Omar stood between the Black Stone Corner and the Kaaba door and then put his chest, his face, his arms and his hands with both palms stretched on the wall of the Kaaba and said: "I saw the Prophet in Mecca doing the same thing".
Imam Ahmad and Abu Dawood reported from Abdur Rahman bin Safwan that he said: "When the Apostle of Allah (peace be upon him) conquered Mecca, I said (to myself): I shall put on my clothes, and my house lay on the way, I shall watch how the Apostle of Allah (peace be upon him) behaves. So I went out. I saw that the Prophet (peace be upon him) and his Companions had come out from the Ka'bah and embraced the House (the Ka'bah) from its entrance (al-Bab) to al-Hatim. They placed their cheek on the House (the Ka'bah) while the Apostle of Allah (peace be upon him) was amongst them".
Allah knows best.