What is the legal ruling regarding opening a door in the mosque specifically for the Imam's entry and exit to his house adjacent to the mosque?

There is no impediment to opening a door through which the Imam enters and exits the mosque, even if his house is adjacent to the mosque. This is analogous to the chamber of Aisha - may Allah be pleased with her - which was adjacent to the mosque and had a door through which the Prophet - peace and blessings be upon him - would enter. In Sahih al-Bukhari, Aisha said: "The Prophet - peace and blessings be upon him - would put his head into my room while he was in the mosque, and I would comb his hair. He would not enter the house except for a need if he was in i'tikaf." In another narration: "She would comb the hair of the Prophet - peace and blessings be upon him - while she was menstruating and he was in i'tikaf in the mosque, and she was in her chamber; he would extend his head to her."