Hi,
1. This email should only be sent to super users (in fact, only to the email addresses you entered in the "admin notification email address" setting of the HikaShop configuration).
However, there was a bug with some payment methods in the past regarding that. So maybe it's related to this as you apparently have an old version of HikaShop. Updating might be the way to go here.
2. I'm not sure which emails you're talking about here. The admin is not supposed to get the "order creation notification" nor the "order status notification" which are both sent to the customer.
Are you saying that your customer is getting one and not the other ? Or you ? Or are you talking about the "order admin notification" (sent to the admin when the order is created) and the "payment notification" (sent to the admin when the order is confirmed) ?
Are the orders with that issue confirmed properly ? Do you see the emails in the Customers>Emails history menu which logs all the emails sent by HikaShop ?
3. and 4. I would recommend to check what I mentioned above
5. Check the "Clean cart when order is" setting in the HikaShop configuration and make sure it's set to "confirmed". If it's set to "created", then that would explain the issue as in that case, the cart is cleared before the customer is redirected to PayPal, so if you come back to the website before paying, the cart is already not there anymore.