Google-sheets – How to use user-defined functions for custom formatting

The script below will perform a summation throughout tabs, taking into account different strings, as shown in your question.

Code

function mySum(tabRange, strings) {
  // small error handling
  if(typeof tabRange === 'string') {
    throw 'Multiple tabs are needed. Otherwise use SUM';
    return
  }  

  // get active spreasdsheet and create empty arrays
  var ss = SpreadsheetApp.getActive(), aStr = [], output = [];

  // push first result in array to control different strings
  aStr.push(strings[0][0], 0);

  // iterate through tabs
  for(var i = 0, iLen = tabRange.length; i < iLen; i++) {

    // get all data (at once) from tab
    var data = ss.getSheetByName(tabRange[i][0]).getDataRange().getValues();  

    // iterate through strings
    for(var k = 0, kLen = strings.length; k < kLen; k++) {

      // create and set the summation per string
      var strSum = 0;

      // iterate through the data
      for(var j = 0, jLen = data.length; j < jLen; j++) {

        // if a match is found, then add to the strSum
        if(data[j][1] === strings[k][0]) {
          strSum += data[j][0];
        }
      }

      // find index in array that matches a string
      var index = aStr.indexOf(strings[k][0]);

      // if found, then add to existing value
      // else, create a new entry for that string
      // so that it will be found the next time
      if(index !== -1) {
        aStr[index+1] += strSum;
      } else {
        aStr.push(strings[k][0], strSum);
      }
    }
  }

  // get rid off the strings  
  for(var l = 1, lLen = aStr.length; l < lLen; l += 2) {
    output.push(aStr[l]);
  }

  // show the outcome  
  return output; 
}

Screenshot

Example

I've created an example file for you: Sum of dynamic number of values derived from fomula

Answering my own question:

=-$A$1 in a less than built-in condition

The = sign in front of the expression makes it an expression. Without it Google Sheets does weird stuff.