On the PHP side, the html of the table is composed with the labels that are there when you enter.
But when you add a label, the html of the added row is built by the javascript, and is different from the one built by PHP.
That is not consistent and is a problem.

In my browser, and I suppose that in many, the "Add" button is simply a <button>. Since it doesn't have type="button", it behaves like a form submit. That is, the click handler adds a row and then the form is also submitted.

The screen should function as the subquestions/answers part of the question editor. That is, every time you press Add it should make a query to the server to fetch the html to add.