Can I get rid of the number_of_uploads parameter on the shopping_cart URL?

[hica]

It seems you need to take some coding.
As far as I know, the number_of_uploads within URL is generated by includes\modules\attributes.php
If you use Zen Cart 1.3.8a, you may find at the end of the file:

$_GET['number_of_uploads'] = $number_of_uploads;

The most simple solution to your task might be:

if($number_of_uploads > 0) {
     $_GET['number_of_uploads'] = $number_of_uploads;
     ...
}