What is the ruling on a mosque neighbor's continuous use of the mosque's ablution facilities for Wudu, with the resultant waste of water and soiling of the place, and his coming to the mosque without Wudu?

It is better for the worshipper to perform ablution (wudu) at home before going to the mosque. There is no harm in performing ablution at the mosque, but doing so misses the virtue of going to the mosque already purified. Performing ablution at home is even more تأكد (emphasized/confirmed) if the water at the mosque is scarce, so as not to crowd his brothers and to make space for them. This is out of gentleness and ease towards Muslims. The Prophet, peace and blessings be upon him, used to forgo some of his preferences out of gentleness towards Muslims. Allah helps His servant as long as the servant is helping his brother.