You're not naming any of your elements to PHP doesn't know what to do with it.
For your submit button:
<button class="submit" name="submit" type="submit">Submit!</button>
For other things you need to go into the tag and add the name to it like I did above there.
If I did this with a name "test1":
<input type="text" name="test1" />
and sent it, like you're doing through POST, it could be read in PHP as:
$_POST['test1']