What is the ruling on the validity of the i'tikaf of one who intended to perform i'tikaf for ten days in Ramadan, but excluded the time of Isha and Tarawih prayers because he is an Imam at another mosque?

The basic principle is that one who is in I'tikaf (spiritual retreat) must remain in their place of I'tikaf and not exit it before the end of the period, except for necessities. As for exiting for something that can be avoided, it invalidates the I'tikaf, unless exiting was stipulated as a condition. So, if the person in I'tikaf stipulates exiting to lead the people in the 'Isha and Taraweeh prayers in another mosque, then their I'tikaf is valid and is not invalidated by exiting for what was stipulated from the beginning.