ISO weeks begin on Monday. Week 1 is the week that contains the first Thursday of the year. A late December date can belong to week 1 of the next ISO year. A January date can still belong to the last ISO week of the previous year. The monthly template can print those week numbers in a gutter if you turn them on.
The weekday name is the civil day in the Gregorian calendar. It does not depend on whether your printed grid starts on Sunday or Monday. Only the column position changes when you switch the start.