Full example (missing some numbers etc..)
can point whereveryou want.. and it stays there.
Creates the keyboard dynamically,only one eventlistener
<!doctype html>
<html>
<head>
<meta charset="utf-8">
<title>keyboard</title>
<style>
body>div{
 clear:both;
 overflow:auto;
 border:2px solid grey;
}
body>div>div{
 width:64px;line-height:64px;float:left;
 border:1px solid grey;
 text-align:center;
}
</style>
<script>
(function(W){
 var D,K,I,pos=0;
 function init(){
  D=W.document;
  I=document.createElement('input');
  document.body.appendChild(I);
  K=D.createElement('div');
  K.id="k";
  K.addEventListener('click',h,false);
  var L='a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z'.split(','),
  l=L.length;
  for(var a=0;a<l;a++){
   K.appendChild(document.createElement('div')).innerText=L[a];
  }
  document.body.appendChild(K);
 }
 function h(e){  
  if(e.target.parentNode.id=='k'){
   pos=(I.selectionStart?I.selectionStart:pos?pos:0);
   var end=I.selectionEnd?I.selectionEnd:pos;
   I.value=I.value.substr(0,pos)+
   e.target.innerText+
   I.value.substr(end);
   I.focus();
   pos++
   I.selectionStart=pos;
   I.selectionEnd=pos;
  }
 }
 W.addEventListener('load',init,false);
})(window)
</script>
</head>
<body>
</body>
</html>
ps.: I tested in Chrome.
EDIT
the only thing that doesnot work is if you select a text and write before deleting it it starts where te selection starts and leaves yor other selected letters where they are.
EDIT 2 everything you expect works