What is the covenant between Omar and the Christians in Jerusalem, and is it possible to protect churches? And is the mentioned report authentic?

It is weak that Umar ibn al-Khattab made peace with the people of Aelia (Jerusalem) and wrote the treaty for them in Jabiya, because Sayf ibn Umar al-Tamimi, the narrator of the report, is a discredited hadith narrator (matruk al-hadith), and Abu Uthman and Abu Harithah are unknown (majhul).

Jerusalem was indeed conquered peacefully, and it is permissible to accept the Christians' condition that existing churches at the time of the peace treaty should not be demolished. This is because the terms of peace are agreed upon with its people according to what the Muslim ruler deems to serve the best interest. Just as the Messenger of Allah, peace and blessings be upon him, made peace with the people of Najran and did not stipulate that they should not establish a new church or monastery.