Is Friday prayer performed at any time, or is its time from before noon until the end of Dhuhr time, and what is the authenticity of the hadith that my shaykh used as evidence for that, which states that the Prophet (peace be upon him) continued to address the people through Dhuhr, Asr, Maghrib, and Isha?

Scholars are in agreement that time is a condition for the validity of the Jumu'ah prayer, and that it cannot be made up in its original form. The majority of scholars hold that its time is the same as the Dhuhr prayer, both in beginning and end. The Hanbalis alone permitted performing it one hour before Zawwal (zenith), and some of them (permitted it) from the sun's rising high, like the Eid prayer. The Malikis alone extended its time until sunset, but this opinion is anomalous and has been strongly weakened by scholars, as the majority of scholars are in agreement that its last time is the last time of the Dhuhr prayer. No scholar has ever said that the Jumu'ah prayer can be performed at any time of the day. As for some people's reliance on the hadith about the Prophet, peace and blessings be upon him, delivering a sermon throughout the day, this was not the Jumu'ah sermon; rather, it was a sermon of exhortation and teaching, as evidenced by his praying Dhuhr and Asr within it.