Within the default Wordpress template, there's this in header.php.

<body <?php body_class(); ?>>

This echos some helpful hooks for CSS and JavaScript. Some example output is

<body class="page page-id-45 page-template page-template-gallery-php">

Is it feasible, inside the header.php even, to include my very own classes additionally as to the can there be?

Particularly, Let me perform a quick strstr on $_SERVER['HTTP_USER_AGENT'] and give a class when the system is an iDevice.

I figure I possibly could get it done with JavaScript, but nonetheless having the ability to understand how to add my very own classes could be helpful nonetheless.

In my opinion you can easily pass the additional class-title like a parameter, i.e.

<body <?php body_class('myclassname'); ?>>

...which appears to become confirmed here.