Changing sth. with Javascript(using Gm and jQuery)in the website when an option is selected

Answer 1

You can use the .selectedIndex property to get which one's selected, like this:

if(document.getElementById("idname").selectedIndex === 0) { //option 1
  //perform the script
}

It's a 0-based index, so 0 is Option 1 in your example markup, you can test/play with it here.

---

I'm unclear from the question, but if you want this on the change event, it'd look like this:

document.getElementById("idname").onchange = function() {
  if(this.selectedIndex === 0) {
    //perform the script
  }
};

You cant test it here.

Answer 2

Update:

Ok, if I understand the revised question, the code now works as intended except when option 1 starts as selected. (PS, the id's of the given code should be edited to match up.)

If that's true, then just change this line:

$("#tabelle").hide();

.
To this:

if ($("#option-type")[0].selectedIndex == 0 )
    $("#tabelle").show();
else
    $("#tabelle").hide();

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In Greasemonkey, you can't set event handlers that way due to sandbox protection. See Common Pitfalls in Greasemonkey.
Also, with jQuery, there are easier ways to select that element.

Something like: $("#idname option:first").select (YourEventHandler) should work.

Where:

function YourEventHandler (Event)  
{
    //YOUR CODE HERE
    //Note: Event is a variable passed automatically to all event handlers, you often just ignore it.
}

Handy jQuery reference.