Hi,
1. While there is no "proper" record, if you didn't disable the "email history" plugin of HikaShop in the Joomla plugins manager, then in the Customers>Emails history, you can see all the emails sent by HikaShop, including the "contact request" emails.
2. There is no mechanism to keep the contact mechanism within the order page itself. It would require changing the whole system.
3. The user cannot see its contact history. As the admin, you can't directly see them from the order details page. But you can access them via the Customers>Emails history menu.