Google-sheets – Add values between two dates in Google Sheets

If you did not insist on array-based formula, then

sumifs(C:C, A:A, A2, B:B, "<="&B2)

entered in D2 and copied around the D column, would do the job.

I don't think that an arrayformula-based solution exists, because arrayformula runs through an array once, performing some element-wise operations. You want to compare each element with all others, which is one step further in complexity (a double for loop).

But you can use a custom function, something like =runningtotal(A2:A, B2:B, C2:C) which processes the columns and returns an array of running totals. The script implementing this function is below; add it via Tools > Script Editor. The logic is quite straightforward: as said above, it's a double for loop.

function runningTotal(names, dates, amounts) {
  var sum, totals = [], n = names.length;
  if (dates.length != n || amounts.length != n) {
    return 'Error: need three columns of equal length'; 
  }
  for (var i = 0; i < n; i++) {
    if (names[i][0]) {
      sum = 0;
      for (var j = 0; j < n; j++) {
        if (names[j][0] == names[i][0] && dates[j][0] <= dates[i][0]) {
          sum = sum + amounts[j][0];
        }
      }
    }
    else {
      sum = '';
    }
    totals.push([sum]);
  }
  return totals;
}

You can do a count for each month by transforming an array formula to only count based on that month:

=COUNTIF(ARRAYFORMULA(TEXT('Canada Twitter Raw Data'!A2:A,"MM")),"01")

change the "01" for each month so february would look like this:

=COUNTIF(ARRAYFORMULA(TEXT('Canada Twitter Raw Data'!A2:A,"MM")),"02")