$random = $wpdb->get_var("SELECT ID FROM $wpdb->posts WHERE post_type = 'furniture' AND post_status = 'publish' ORDER BY rand() LIMIT 1");

For instance, if I wish to specify a width and height, where would I input this?

If you wish to fetch width and height out of your SQL Table using above structure so that you can use

$random = $wpdb->get_row("SELECT ID,width,height FROM $wpdb->posts WHERE post_type = 'furniture' AND post_status = 'publish' ORDER BY rand() LIMIT 1");
echo $random->id;
echo $random->width;
echo $random->height;

For this type of question there's just one answer.

$width = 0; // Enter width
$height = 0; // Enter height
$random = $wpdb->get_var("SELECT ID FROM $wpdb->posts WHERE post_type = 'furniture' AND post_status = 'publish' AND width = {$width} AND height={$height} ORDER BY rand() LIMIT 1");