Count IF Average, Median , SD and Range

I think you will need to use Arrayed formulas. Therefore you want to first create dynamic defined names so that Excel is not trying to do an array on an entire column and you don't need to reset it every time you add data. In "Name Manager",

Dates =Data!$C$2:INDEX(Data!$C:$C, COUNT(Data!$C:$C)+1)
Data1 =Data!$H$2:INDEX(Data!$H:$H, COUNT(Data!$C:$C)+1)
Data2 =Data!$I$2:INDEX(Data!$I:$I, COUNT(Data!$C:$C)+1)

To get Average, you can use a straight up AVERAGEIFS (formulas from row 2)
=AVERAGEIFS(Data1,Dates,">="&A2,Dates, "<="&EOMONTH(A2,0))

For the others, you'll need to use Arrays
...confirmed by pressing CTRL+SHIFT+ENTER to activate the array, not just ENTER. You will know the array is active when you see curly braces { } appear around your formula. If you do not CTRL+SHIFT+ENTER you will get an error or a clearly incorrect answer. Press F2 on that cell and try again.

Median Data1
=MEDIAN(IF(Dates>=A2,IF(Dates<=EOMONTH(A2,0),Data1)))
Standard Dev Data1
=STDEV(IF(Dates>=A2,IF(Dates <= EOMONTH(A2,0),Data1)))
Range Data1
=MAX(IF(Dates>=A2, IF(Dates<=EOMONTH(A2,0),Data1)))-MIN(IF(Dates>=A2, IF(Dates<=EOMONTH(A2,0),Data1)))
See attached.
Does that work for you?

Create a dynamic named range for Col A (I used IDName) and
You will just add another "IF Statement" criteria to each formula, for example, for Median

=MEDIAN(IF(Dates>=A17,IF(Dates<=EOMONTH(A2,0),IF(IDName="TT",Data1))))

With your limited data, you get errors for some dates. You could wrap an IFERROR around your formulas to remove those errors.