What is the ruling regarding some worshippers standing next to the Imam without the first row being complete, even though the mosque is not full? And which of the two rows is considered the first row: the complete one, or the one next to the Imam?

The Sunnah is for the congregants to stand behind the Imam, unless there is a need, such as limited space. Otherwise, their prayer is valid even if they contravene the Sunnah. The first row is the one directly behind the Imam. Anyone who prays next to the Imam is not considered to be in the first row. It is permissible to pray to his right or on his sides when there is overcrowding.