Is it obligatory for the Imam to call the Adhan for Isha prayer immediately after finishing Maghrib prayer when combining the two prayers due to rain, or is it permissible for him to recite the prescribed supplications between the two prayers? And what are the required supplications if he proceeds directly to Isha?

For the validity of Jam' al-Taqdim (combining prayers by performing the second prayer at the time of the first), it is a condition to perform the two prayers in immediate succession (muwalat), such that there is no significant interval separating them. A slight interval does not cause harm, and the definition of "slight" is determined by common practice (urf). It is preferable that the worshipper does not engage in anything between the two prayers; rather, they should combine them without any interruption, whether by dhikr (remembrance of Allah) or anything else. If the call to prayer (adhan) for Ishaa is made after Maghrib, it should be done inside the mosque in a low voice, and no dhikr or other recitations should precede it.