Multiple Taxonomy post query with exclusion

If I understood you correctly, an array of terms either slug or term_id, for example, depending on field to be exact.

$args = array(
    'post_type' => 'a_post_type',
    'tax_query' => array(
        'relation' => 'AND',
        array(
            'taxonomy' => 'tax_one',
            'field'    => 'slug',
            'terms'    => array( 'action', 'comedy' ),
        ),
        array(
            'taxonomy' => 'tax_two',
            'field'    => 'term_id',
            'terms'    => array( 103, 115, 206 ),
            'operator' => 'NOT IN',
        ),
    ),
);

See Codex: WP_Query

For the form, in short, make name an array basically, e.g.:

<form method="post" name="en_tax" >
<input type="checkbox" name="post_variable[]" value="value 1">
<input type="checkbox" name="post_variable[]" value="value 2">
<input type="submit" />
</form>

You then can get the variable $_POST[ 'post_variable' ], which is an array of values.