How can the sighting of the crescent moon be followed for Islamic legal purposes outside of Ramadan and Hajj, and does the date announced on the website rely on Islamic legal sighting and not on calendars?

The White Days are the thirteenth, fourteenth, and fifteenth of the month. The start of the month is determined by sighting the crescent moon, or by relying on the calendar with the strong presumption [of its accuracy]. An error in determining one of these days does not matter, because the fasting person is excused. Fasting three days of every month is a Sunnah, and it is preferable for them to be on the White Days, but there is no harm in fasting them at any time of the month if it is difficult to fast them on the White Days.