The "picture" is a graphical hyperlink. Messages does that by design. It's functionally identical to the non-graphical address. It should open in the applicable app when you tap on it. I believe there's a way to copy and paste the non-graphical address but that would be a waste of time if not counter-productive.
When Internet isn't fully iMessage-available and you go to send a message that would otherwise go as iMessage, Messages will turn around and send the message as SMS or MMS "Text Message" as long as Send as SMS is enabled in Messages settings on the iPhone.
Your VPN provider probably has iMessage blocked and you're unlikely to get the provider to unblock it. If you have Internet access at all via the VPN, that would be via proxy settings wherein iMessage would be blocked. If Send All Traffic is enabled in the VPN settings, then you would have to disconnect from the VPN to use iMessage via Wi-Fi or cellular data. Otherwise, disabling that setting might allow iMessage to divert to 4G data even while you're connected to the VPN. If the VPN appears by virtue of a configuration profile from the provider, you could try to get the settings from the provider and delete the config profile and set up the VPN yourself with Send All Traffic disabled.