You need to alter this code:
$("#selectestate").bind("change",function(){
to this:
$("#selectestate select").bind("change",function(){
You are asking for the value of the <td id="selectstate"> ... which is of course null.  #selectstate select loads the actual <select> element that you are looking for, which will then enable you to get its value.
You will also want to change $("#selectcity") to $("#selectcity select") to avoid the same problem in your anonymous function.  
Finally, your alert is behaving as expected. {} defines an object in Javascript.  So you are alerting an object containing a single attribute.  Just alert($(this).val());// Should show the value
If it still fails it's because either:
A) Your URL is wrong
or
B) There is something wrong with the php function called by results/ajaxcity
EDIT:
Smacks head: I should have caught this.  In your code you have this function:
1.    function getState(){
2.    $("#selectestate select").bind("change",function(){
3.    
4.        $("#selectcity select").load("results/ajaxcity", {stat: $(this).val()} ); 
5.        alert($(this).val());
6.        });
7.    return false;
8.    }
Your generated HTML looks something like this:
<select name="state" id="selectstate" onChange="getState();">
THE PROBLEM: When you use the <select> for the first time your function is called and jQuery binds an anonymous function to the change event for select (line #2) that will be executed every time the change event fires from this select from now on.  Every time you select the dropdown a new anonymous function is bound by line #2 and all of the functions that are currently bound to the dropdown are executed. (So if you use the dropdown N times the "change" function will fire N-1 times.)
THE SOLUTION: Use $(document).ready() to bind your function.  Your restructured code will look like this:
<script type="text/javascript">
function getState(element){
    $("#selectcity select").load("results/ajaxcity", {stat: $(element).val()} );
    alert($(element).val());
    return false;
}
</script>
//... snip ...
<?php $js = 'id="selectstate"';?>
//... snip ...
<script type="text/javascript">
$(document).ready(function() {
    $("#selectestate select").bind("change", getState);
});
</script>