Country data archiving script using Google Spreadsheet
You have a lot of repeated code when creating the data array. Note that the names of each cell value are ultimately irrelevant. That part could also be expressed as:

var columns = ["A" // office
              ,"B" // title
              ,"C" // imdbId
              ,"D" // channel
              ,"E" // type
              ,"F" // added
              ,"G" // deadline
              ,"H" // airing
              ,"I" // link
              ,"J" // picture
              ,"K" // comment
              ,"L" // portId
              ,"M" // status
              ];
var data = columns.map(function (col) {
  return source_sheet.getRange(col + i).getValue();
});


Less redundant code, same level of documentation.

Code like

if (...) {
  ...;
  i++;
} else {
  i++;
}


is usually better written as:

if (...) {
  ...;
}
i++;


However, we now have

var i = 2;
while (i &lt;= lastRow) {
  ...;
  i++;
}


which is a more complicated formulation of

for (var i = 2; i &lt;= lastRow; i++) {
  ...;
}


Similarly,

rowsToBeDeleted.reverse();
for (var j = 0; j &lt; rowsToBeDeleted.length; j++) 
{
    source_sheet.deleteRow(rowsToBeDeleted[j]);
}


is the same as:

for (var i = rowsToBeDeleted.length - 1; i &gt;= 0; i--) {
  source_sheet.deleteRow(rowsToBeDeleted[i]);
}


If I see that correctly, your two function only differ in the strings croatian_backup vs. serbian_backup and CRO vs. SER. Instead of copy-pasting that code, take those strings from the function parameters. Now we have a general archive function:

function archive(outputSheet, languageCode) { ... }


If you need two seperate funtions that don't take any arguments, we can use the currying technique (also known as partial application) to pre-fill the arguments:

function CroatianArchive() {
  return archive("croatian_backup", "CRO");
}


Some of your names use underscores, other capitalization to separate words:

source_range
lastRow


You should settle for one style – Javascript tends to prefer capitalization rather than underscores.