Google-sheets – quickly sort Google Sheet via buttons

I don't think rowHeight can help you here, because it refers to the entire row, not to any particular cell in it. Sorting each column by RowHeight would just rearrange the rows without making your spreadsheet more compact.

Here is the script that sorts each column by the length of cell content, ascending. The first row (headers) is left in place; empty cells are ignored.

function myFunction() {
   var ss = SpreadsheetApp.getActiveSpreadsheet();
   var sheet = ss.getActiveSheet();
   var range = sheet.getDataRange();
   var width = range.getWidth();
   var height = range.getHeight();
   var column, i, j, content;
   for (j = 1; j <= width; j++) {
     column = [];
     for (i = 2; i <= height; i++) {
       content = range.getCell(i,j).getValue();
       if (content) {
         column.push(content);
       }
     }
     column.sort(function (a,b) {return a.length - b.length;});
     for (i = 0; i < column.length; i++) {
       range.getCell(i+2,j).setValue(column[i]);
     }
     for (i = column.length; i < height-1; i++) {
       range.getCell(i+2,j).setValue("");
     }
  }
}

But in practice, this particular sort does not help all that much, because (as you noticed) the longer texts can still match up against shorter texts. Here is another sort, somewhat along the lines of what you mentioned: after sorting, each column is dropped down so that all the longest entries are in the same row. This will typically result in some empty cells at the top.

function myFunction2() {
   var ss = SpreadsheetApp.getActiveSpreadsheet();
   var sheet = ss.getActiveSheet();
   var range = sheet.getDataRange();
   var width = range.getWidth();
   var height = range.getHeight();
   var column, i, j, content;
   for (j = 1; j <= width; j++) {
     column = [];
     for (i = 2; i <= height; i++) {
       content = range.getCell(i,j).getValue();
       if (content) {
         column.push(content);
       }
     }
     column.sort(function (a,b) {return a.length - b.length;});
     for (i = 0; i < column.length; i++) {
       range.getCell(height-column.length+i+1,j).setValue(column[i]);
     }
     for (i = 2; i <= height-column.length; i++) {
       range.getCell(i,j).setValue("");
     }
  }
}

You can achieve this by using an array of importrange() functions. This cannot import the formatting information, which is not possible without Apps Script, but it will pull in multiple sheets and aggregate them into one list, which you can then sort with an additional formula. Keep in mind that each imported sheet will have to have the same number of columns. For example, if you have four sheets to import with 4 columns each, the import formula will be as follows:

={
importrange("url_to_sheet_1", "Approved!$A:$D");
importrange("url_to_sheet_2", "Approved!$A:$D");
importrange("url_to_sheet_3", "Approved!$A:$D");
importrange("url_to_sheet_4", "Approved!$A:$D")
}