First, as I said in a comment already, mysql_
functions are deprecated and may stop working in later PHP releases, so you should be using PDO for new development.
But doing what you want is as simple as using javascript to submit the form on change of the selection:
<form name='formname' action='theSamePage.php' method='get'>
<select name='whatever' onChange='document.formname.submit();'>
And getting the WHERE clause right in your SQL. If the previous selection is not set, you don't even run the SQL or print the next dropdown's options, and it does, you do.
Also, as you loop through the options from the db to print them out, check if the current option from the db equals the one received in the request (if its set) and if it does, instead of printing just <option value='x'>x</option>
print <option value='x' selected='true'>x</option>
to keep the option selected when the page refreshes.
Take the selected out of <option selected>Select Error/Problem</option>
since that will make the blank option always selected. If no selection is chosen yet, the first one will always be seen as selected anyway by the way the browser works.