Exclude Current and Sticky Post

Whenever an array of arguments is a function parameter in a WP core function it is parsed via wp_parse_args and almost always extracted into single variables.
I.e. no, you cannot use the same argument twice.

What you want to do is something like this:

$exclude = get_option( 'sticky_posts' );
$exclude[] = get_the_ID();

$args = array(
    'post_type' => 'post',
    'post_status' => 'publish',
    'order' => 'DESC',
    'orderby' => 'date',
    'posts_per_page' => 3,
    'post__not_in' => $exclude
);

As an aside, you were also missing a comma.