You cannot talk to a backing bean directly from JavaScript. That's what AJAX is for.
You can probably set up custom styles for the PrimeFaces date picker control that will make it look like Bootstrap's date picker, but you might want to look at BootFaces, which is specifically designed to make
JSF look and feel more like Bootstrap.
The secret of how to be miserable is to constantly expect things are going to happen the way that they are "supposed" to happen.
You can have faith, which carries the understanding that you may be disappointed. Then there's being a willfully-blind idiot, which virtually guarantees it.