Is it permissible for the Imam to deliver a sermon to the people in the mosque between the Fajr adhan and the iqamah, or is it obligatory to recite the Qur’an and engage in tasbih, and is this considered a disturbance to those praying and reciting the Qur’an?

There is no harm for the Imam to deliver a lesson between the Adhan and the Iqamah due to people's need for education and admonition. However, it should not be a perpetual practice so that people do not get bored. Indeed, Ibn Mas'ud used to alternate in admonishing his companions, disliking that they would tire. Seeking knowledge is better than supererogatory acts of worship. And whoever wishes to recite the Qur'an may do so, but it is not obligatory.