Second order no address details in adm. email

The admin notification is the same for the first and next emails.
That fact that the shipping address is not displayed is because of a bug.
You should replace the line:
if(!empty($data->cart->has_shipping) && !empty($data->cart->shipping_address)){
by:
if(!empty($data->cart->order_shipping_id) && !empty($data->cart->shipping_address)){

in the order admin notification.

Would it be possible to have a back end access to your website in order to check out the problem because I'm out of ideas ?

We got them. As it's already late around here, we'll look at the problem tomorrow.

Hi,

I'm sorry but I just tried to go on your back end and the account you provided did not enable me to log in. Could you please double check the credentials ?

I was able to login on your back end now.
The problem is that you don't have the view "address" on your checkout workflow. That's the view responsible for the address selection during the checkout. If you don't have it, the system does not select any address for the order. Which is why you don't see any address in the emails for users with an account.
For users creating a new account, the address information is asked on the login view and is automatically selected when the account is created so that's why it works for new accounts and not existing accounts.