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For the first question, I'm assuming that C4:AG4 (and C12:AG12 and C19:AG19, etc) hold just a number like 1, 2, 3... 31 for the day of the month.
Put this formula into C5:
=WEEKDAY(DATE(TheYear,MONTH($B4),C4))
and then fill it out to column AG and then format all of them Custom with dddd and they should then show Sunday, Monday, Tuesday, etc.
Now for the highlighting of weekends. Select all of the C4:AG4 cells and use conditional formatting and create a new rule based on a formula. Here is the formula to enter:
=OR(WEEKDAY(C5)=1,WEEKDAY(C5)=7)
and choose the formatting to highlight those cells.
You can copy the formulas and formatting down into the other cells in the months that follow on the sheet.
Hope this helps