Short answer

There is no need of scripts. Instead use the built-in arrays handling features of Google Sheets.

Explanation

Google Sheets can handle arrays and automatically expand them. There are some functions like SUMPRODUCT, MMULT, FILTER and QUERY that return arrays but other functions require to be used together with ARRAYFORMULA.

Not all the functions could work with ARRAYFORMULA or will return an array of values. Below are the array formulas that should work for the cases exposed in the question.

Examples

Add the following formulas to row 2.

=ArrayFormula(WEEKNUM(FILTER(C2:C,ISNUMBER(C2:C)))-6)
=ArrayFormula(FILTER(B2:B&F2:F,LEN(B2:B)) 

Short answer

From the question:

edit: where trying to use an implementation of the script where the result satisfies a condition, I'm not certain but I think the sheet "views" the formula input as giving a result, albeit blank? So not a value of "0"

Instead of something like

values[i][0] > 0

you should use

values[i][0] != ''

Explanation

Part 1: Misconception about blank cells and empty strings.

Your formula doesn't return a blank, it returns a empty string "". A blank cell is coerced to 0 always but a empty string doesn't.

Example:
If A1 has the following formula ="", =ISNUMBER(A1) returns FALSE but =A1+1 returns 1.

By the other hand, on Google Apps Script a blank cell value is handled as an empty string, not as 0 (zero).

Google Apps Script is based on JavaScript no on the "spreadsheet thinking" so there are several things like the default value of a blank cell that are handle in a different way than it's on Google Sheets.

Part 2: One practice for Google Apps Script troubleshooting

In case of doubts, a general practice is to create simple test functions that return the result to the log but in the case of Google Sheets you could use a custom function if you feel more comfortable with the Google Sheets UI than with the Script Editor.

To check the value that Google Apps Script got from an empty string you could use something like the following custom function:

/**
 * Returns a 2D array 
 *
 * @param {"A1"} input Cell reference string
 * @customfunction
 */
function valueDataType(input){
  var ss = SpreadsheetApp.getActiveSpreadsheet();
  var range = ss.getRange(input);
  var value = range.getValue();
  var output = [  
     ['Value: ' + value],
     ['Data type: ' + typeof value]
  ];
  return output;
}