Is this Hadith used as proof for the desirability of one who enters the mosque on Friday to pray continuously until the Imam enters, and was this action authentically reported from the Companions during the time of the Prophet, peace and blessings be upon him, or after him?

The aforementioned Hadith in the question was narrated by Muslim, and it encourages supererogatory prayers (Nafl) before the Friday prayer in general, without specification, and that there is no limit to absolute supererogatory prayers.

Jurists have deduced that voluntary prayer ceases when the Imam sits on the pulpit. Voluntary prayer stops when the Imam sits on the pulpit, so no one prays except for the one entering the mosque, who may pray two short rak'ahs as a greeting to the mosque.

The majority of Imams agree that there is no prescribed Sunnah prayer before Friday prayer that is fixed by a specific time or a set number of rak'ahs.