PHP can generate the form's HTML and it can handle the data from a submitted form, but it doesn't "manage" forms or do anything on the client side, so I'm not sure what you're talking about there.
Now, for the first error in the w3 validator:
"document type does not allow element "form" here..."
And then it complains (vaguely as it sometimes does) about incorrect containers.
Here's the section of code:
<form method="post" action="">
<!-- stuff here -->
Drop those paragraph tags and revalidate.