You can have the form with the select lists post to a new page with the map on it or the same page, it does not matter. If you have it post to the same page and you would like to retain the values the user selected you can set the selected value of the dropdown in the property inspector. Just choose the same form element from that form in the bindings when setting the initial value.
If you want to make the map display conditional on values being posted to the page you wrap the map div with an if statement like this:
<?php if($_SERVER["REQUEST_METHOD"] == "POST"){ ?>
the map div goes here
<?php } ?>
As for the problem with the place holder box on the server have you ensured that you have uploaded all of the files for the map? You must make sure that all the files in the google_javascript folder on present on the server. Also, if you are testing locally and have a good result you may need to update your api key for the live server.
