Is it better for the regular imam to step forward to lead the prayer, or should he let the muezzin lead, if the regular imam enters while people are waiting for the prayer, and the muezzin has already stepped forward to lead them?

The regular Imam has the foremost right to lead the prayer, and no one should step forward ahead of him except with his permission, based on the Prophet's (peace be upon him) saying: "Do not lead a man in prayer in his own home or where he has authority, nor sit in his place of honor in his house unless he gives you permission or with his permission." If the regular Imam is late, the people of the mosque may pray before him if he is absent or far away and his presence is not expected, or if time is short. However, if he is present or nearby and there is ample time, then he should be awaited.