Autoplaying the video should be possible by appending this to the video URL:
?autoplay=1
To dynamically display the navigation buttons, I'd add an ID to the iframe element like this:
id="myvimeo"
Then add this script to the page:
<script src="https://player.vimeo.com/api/player.js"></script>
<script>
$(document).ready(function(){
$('#next_button, #previous_button').hide();
var iframe = $('#myvimeo');
var player = new Vimeo.Player(iframe)
player.on('ended', function() {
$('#next_button, #previous_button').show();
});
});
</script>
Disabling picture-in-picture seems to be the default behavior. I don't see that button when I use your HTML.