How to remove "Show row weights" option on File Upload field? I have created a file field with multiple uploads allowed. Is there any way(except CSS) I can remove "Show row weights" link from file upload field? Or is there any way to remove weight drop down entirely from file field?
The first part of your question is regarding removal of link "Show row weights" (or "Hide row weights" if dragging functionality is disabled) in a form where table rows are draggable, so I'm going to answer that.

"Show row weights" string appears in Drupal.tableDrag.prototype.showColumns() method and "Hide row weights" appers in Drupal.tableDrag.prototype.hideColumns() method that is included in tabledrag.js file (see head in Drupal 7.x branch) so you need to override these methods in order to replace or remove these strings without losing row dragging functionality.

Place this code in a .js file that is loaded along with the form that uses tabledrag.js:

(function ($) {
  // Copy hideColumns() method
  var hideColumns = Drupal.tableDrag.prototype.hideColumns;
  Drupal.tableDrag.prototype.hideColumns = function() {
    // Call the original hideColumns() method
    hideColumns.call(this);
    // Remove the 'Show row weights' string
    $('.tabledrag-toggle-weight').text('');
  }

  // Copy showColumns() method
  var showColumns = Drupal.tableDrag.prototype.showColumns;
  Drupal.tableDrag.prototype.showColumns = function () {
    // Call the original showColumns() method
    showColumns.call(this);
    // Remove the 'Hide row weights' string
    $('.tabledrag-toggle-weight').text('');
  }
})(jQuery);