Inserting Fields from A Custom Content Type into Custom Form
There isn't an API for attaching a single field from an entity to a form, there is one to attach all fields. field_attach_form. I wouldn't suggest you doing that do, since it can quickly become unmaintainable.

Instead I would suggest that you simply get the field options, or allowed values as they are called. That is pretty simple as well.

$info = field_info_field('field_machine_name');
$options = $info['settings']['allowed_values'];