The Month and Year table on this template has one column and fourteen rows, and displays only the first two rows.

The values used in conditional highlighting rules rules 1 and 2 for the cells displaying 'dayname-daynum' in the 'CALENDAR' table compare the date in the cell to be highlighted with the first and last dates of the month from the 4th and sixth rows of the Month and Year table. then apply the custom format for the first rule that returns 'true'. If neither rule returns 'true', the cell displays its data in the format set as default for that cell.
To change the 'conditional highlight' to 'blank', the easiest route is to set the text colour to 'white' when the condition in the second rule is met (for dates after the last day of the current month), and, if you also want the date cells to appear empty for the dates before the first of the current month to appear 'blank' , to set the same conditional highlight for that rule.
On your CALENDAR table, you have changed each of the 'notes' cells to a group of five cells, containing several different types of data:
text (T. T2, OFF, MD, ALBARIQI, H)
Duration in two formats ( 4:00, 2:00, 6hr 55m)
and Numbers: (561, 802).
CH rules compare the value in the cell to be highlighted to a fixed value, written into the rule, or to the value in another cell.
The values in each of these cells
Is the data in these cells entered directly into the cell, or is it retrieved from a cell on a different table or calculated using values from another table?
If it is created using a formula, you can wrap the formula in an IF statement that will insert a null string ( "" ), making the cell appear empty, if the date in the cell associated with this data has a date before or after the first or last day of 'this month', and will calculate and return its intended result if the date in that cell is 'within this month'.
If the data in these cells is entered directly, things become more complicated, as you need a 'buddy' cell to contain a comparison value, and a means to make the rule display the value in the main cell so you can see what you're typing while you are editing it, but hiding it, depending on the value in the buddy cell and the date in the associated date cell when you are not editing the main cell.
More to come, but I'll leave it at that for now.
Regards,
Barry