Is what one of the scholars on Iqra TV mentioned about the Messenger, peace and blessings be upon him, performing prayer in a Companion's house without the adhan, and his saying "the mosque's adhan suffices for that," correct? And is it permissible for a person to perform prayer in his home without the adhan?

It is not obligatory for someone praying at home to recite the Adhan (call to prayer), as the Adhan is a communal obligation (fard kifaya). One Adhan from the mosque that reaches the people of the town is sufficient. As for the Adhan for someone praying at home, it is a recommended act, not an obligatory one. Furthermore, congregational prayer in the mosque is among the greatest rituals of Islam.