How can I remove "Show row weights" as well as the weight column in D7?
If you do not want to display those elements because it confuses users, you could simply hide them with CSS. Use Firebug or something alike to determine the correct classes in order to adapt the code underneath.

#YOUR-FORM-ID .tabledrag-toggle-weight-wrapper,
#YOUR-FORM-ID .field-multiple-drag {
  display: none;
}