(See question above). Does the not drinking water tradition at the start of the Tekufa apply every month? How stringent should one be in keeping this tradition?

It applies only at the four beginning of the Tekufos every year as mentioned above. Rema in Shulchan Aruch (Y.D. 116: 5) quoting Avudraham, Mordechai and others, mentions the above as a most accepted tradition one should comply with. Yet, the practice of not drinking water when the Tekufa begins, is not that commonly kept. Taz (ibid. 4) quotes Even Ezra that it is only a Nichush or superstitious custom. As mentioned on the answer above, it is only four times a year for one hour. Usually those four beginnings are quoted in the Shul calendars. The beginning of the Tekufos also makes a difference in regard to changes on the Tefilos when we pray for rain. Horav Shlomo Miller’s Shlit’a opinion is that one should keep the traditions of his family.