Is it permissible to perform Tarawih prayers on the night of Eid, and what is the evidence for that? And does one receive the reward for fasting the days of Shawwal if they fasted an incomplete number of them?

Tarawih prayer ends with the end of Ramadan (definition) and is not prayed in congregation on the night of Eid. The virtue of fasting the six days of Shawwal is for whoever completes them, unless there is a legitimate excuse that prevents their completion, in which case the full reward is written for him; due to the Prophet's (peace and blessings be upon him) saying: "If a servant falls ill or travels, the like of what he used to do when he was healthy and at home will be written for him."