Is it permissible to perform Tayammum instead of Wudu – even when water is available – if there is difficulty in using the water at the mosque due to its distance from facilities and the lack of suitable places for Wudu, and what is the ruling on prayers performed with Tayammum in this situation?

Allah Almighty has made ablution with water obligatory when it is available and one is able to use it without harm. What is mentioned in the question is not an excuse that permits resorting to Tayammum (definition). You must repeat the prayers that were performed with Tayammum, according to the majority of scholars, because they are prayers lacking the condition of purification, and Tayammum in the presence of water is as if it were nonexistent. However, Sheikh al-Islam Ibn Taymiyyah held the view that no قضآء (repetition) is required when there was ignorance.