The VPN is indeed working in Private Mode too, but the VPN symbol is not showing there. You can try testing for yourself by using a website which checks your IP like https://ipleak.net/ in normal browsing mode and Private Mode.

uBlock Origin only has an option to prevent your local IP from leaking, your public IP will still be visible on this test site: https://diafygi.github.io/webrtc-ips/

99% percent of all DNS leak tests I take I find tons of Google IPs when I'm using the VPN in any of the three countries. Doesn't Google log all their traffic? Opera has said they do not log traffic, but that doesn't prevent Google from doing it, or?

The built in VPN is only available on version 38+, which is only available for Windows 7+ I've heard.

Opera has announced this: "We are currently working hard to implement support for proxying even more of the browser traffic, including WebRTC and plug-ins. Having this functionality built into the browser, instead of as an extension, allows us to catch more situations, such as certificate revocation checks made by the system."
-Source: https://www.opera.com/blogs/news/2016/04/opera-doubling-server-capacity-vpn/

^That means that this tutorial might not be needed in future updates of the Opera browser.

In the meanwhile you can use this tutorial to disable WebRTC in Opera with an extension.
I got the tutorial from: http://www.ghacks.net/2016/04/23/block-opera-vpn-from-leaking-your-ip-address/
But I will simplify it here.

This tutorial is for Windows, I'm not sure about Mac or Linux.

1. (Optional) You can check if Opera leaks your IP here: https://diafygi.github.io/webrtc-ips/

2. Install the "WebRTC Leak Prevent"-add-on: https://addons.opera.com/en/extensions/details/webrtc-leak-prevent/?display=en

3. Open the Opera Menu (top-left corner) and click "Extensions" > "Manage Extensions", or copy/paste "opera://extensions/" in your address bar. Then click "Options" on the "WebRTC Leak Prevent"-extinction.

4. Change the IP Handler Policy to "disable non-proxied UDP (force Proxy)".

5. Select "Prevent WebRTC from using non-proxied UDP"

6. Check if Opera leaks your IP again: https://diafygi.github.io/webrtc-ips/

If you successfully did these steps, your Opera Browser should no longer leak your IP when using the built in VPN. The most important part is changing the IP Handler Policy options to "disable non-proxied UDP (force Proxy)", as that will make sure your IP doesn't leak.