Please clarify the following: Did Adam (sws) ask his forgiveness from Allah by using Prophet Mohammed (sws) as an intermediary (wasīlah)?
The most authentic source regarding the question of what Adam (sws) said in his prayer is the Holy Qur’ān. The incident of the creation of Adam and Eve has been dealt with in various places in the Holy Book. If we study the matter wherever it occurs in the Holy Qur’ān it becomes very clear how Adam expressed his repentance and what he prayed at that time. The incident first occurs in Sūrah Baqarah where it is briefly mentioned:
So Adam learnt some words from his Lord. (2:37)
The verse does not tell what words the Almighty taught him to say in his prayer. In Sūrah A‘rāf (7:22-3) it has been made clear what the Almighty did teach him in this regard:
So by deceit he betrayed them. When they tasted of the tree, their private parts became manifest to them and they began to cover their private parts with the leaves of the Garden. And their Lord called them: ‘Did I not forbid you that tree, and tell you that Satan was your avowed enemy?’ They said: ‘Our Lord! We have wronged our own souls: if you forgive us not and bestow not upon us your mercy, we shall certainly be lost’. (7:22-3)
I think there remains no need of further exploration on the issue. However, you have inquired about a certain point of view, according to which it was the name of the Prophet Muhammad (sws) which Adam (ws) presented as an interceder. None of the reliable books of Āhadīth contain this report. However, Tabarī has given the viewpoint without citing the source or mentioning the names of the scholars who hold this view: 2:37. Therefore, the saying is not authentic in any degree and does not hold water after the explanation given by the Almighty Himself in the Holy Qur’an. What follows is the text of the narrative:
Interpreters have differed on the implication of the word ‘kalimāt’. Ibn Abbās, Hasan, Sa‘īd Ibn Jubayr, Dahhāk and Mujāhid said that it was as He said: ‘Our Lord! We have wronged our ownselves: if you forgive us not and bestow not upon us your mercy, we shall certainly be lost’. And Mujāhid is also reported to have said [that it refers to]: ‘Oh our Lord! There is no god except you, my Lord; I wronged myself so forgive me. Indeed you are the most forgiving, the most merciful’. A group of scholars said: ‘He saw it written on the throne of the Almighty: Muhammad the Messenger of Allah” and he used his name for interceding and this is the meaning of the word ‘Kalimāt’.