Hi,
There is no "receive a copy" option. That's to avoid spam as that capability is often used by spammers to send emails to others (they enter the email address of someone else and their text so that the other gets a spam email sent from your email server (which then lead to your email server being flagged as a spammer).
I would recommend not to add that capability.
If you want to add it, you can edit the "contact" email via the menu System>Emails and add such code in the preload:
if(!empty($data->element->xxx)) {
$mail->cc_email = $data->element->email;
}
where xxx is the column name of your custom contact field.
If you create a custom field of the table "contact" and of the type "ajax file", then the user can add a file to his contact request.
That custom field will be displayed as a download link in the email, not an attachment. To have it as a attachment to the email would require some more complex development but it's not impossible.
It's normal that the ReCaptcha doesn't appear when you're logged in since you already did the captcha when you registered. The goal is to not harass real people with captchas whenever they try to do something on your website.
Regarding the position of the captcha, you have the setting "Display position on contact forms" for the contact form and "Display after" for the registration form. Both are in the settings of the Recaptcha plugin via the joomla plugins manager.