Get current cursor position in a textbox

Question 1

It looks OK apart from the space in your ID attribute, which is not valid, and the fact that you're replacing the value of your input before checking the selection.

function textbox()
{
        var ctl = document.getElementById('Javascript_example');
        var startPos = ctl.selectionStart;
        var endPos = ctl.selectionEnd;
        alert(startPos + ", " + endPos);
}

<input id="Javascript_example" name="one" type="text" value="Javascript example" onclick="textbox()">

Also, if you're supporting IE <= 8 you need to be aware that those browsers do not support selectionStart and selectionEnd.

Question 2

Here's one possible method.

function isMouseInBox(e) {
  var textbox = document.getElementById('textbox');

  // Box position & sizes
  var boxX = textbox.offsetLeft;
  var boxY = textbox.offsetTop;
  var boxWidth = textbox.offsetWidth;
  var boxHeight = textbox.offsetHeight;

  // Mouse position comes from the 'mousemove' event
  var mouseX = e.pageX;
  var mouseY = e.pageY;
  if(mouseX>=boxX && mouseX<=boxX+boxWidth) {
    if(mouseY>=boxY && mouseY<=boxY+boxHeight){
       // Mouse is in the box
       return true;
    }
  }
}

document.addEventListener('mousemove', function(e){
    isMouseInBox(e);
})