Is it correct to say that it is impermissible to touch the Kaaba and Hijr Ismail during tawaf, and that touching is restricted only to the Black Stone and the Yemeni Corner, and is this considered part of the Shafii school of thought?

No statement has been found from any scholar of Islamic law indicating the invalidity of circumambulation (tawaf) for one who touches the Kaaba or the Hijr. The most that has been mentioned regarding this is that it is not prescribed (غير مشروع). The Hanafis and Malikis explicitly stated the detestability (karahah) of touching the two Iraqi and Levantine corners (a minor detestability according to the Hanafis). The Shafi'is said that it is not sunnah to touch them or kiss them, but if one kisses them or any other part of the House, it is not detestable but rather good. The Shafi'i madhhab and the other three madhhabs do not consider tawaf invalid due to touching the Kaaba or the Hijr. It is best to limit oneself to touching the Yemeni Corner and kissing the Black Stone, and to refrain from touching anything else, in imitation of the action of the Prophet, peace and blessings be upon him.