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I assume logging into sites will leave a clear trace?
Any action you perform on a site will probably be logged by that site and leave a trace on the server.
Just viewing a site will probably see an attempt at tracking, that uBlock should handle and prevent.
In the case of the server logs, it's up to the site in question. For LFGSS logs are only retained for 72 hours for the vast majority of data, and only a small amount of non-identifying data is kept beyond that time.
Short answer: Yes.
Better answer: Mostly.
It will run everything over the VPN for the routes that the VPN declares, which is typically the common ports across all non-local IP addresses. Obviously local network traffic remains local, so it should also be obvious that not everything goes via the VPN. It is further possible to configure a VPN to not be used for certain apps, protocols or websites. I actually do this on my Android, because I let Google Play Music download music direct from Google and to allow Cast to work on the local network, whilst everything else goes over the VPN.
Most VPNs and VPN software offers a network kill switch if you want to be at the extremes of things, you can disable networking if the VPN isn't being used for a communication.
As a generalisation though... yes, everything goes over the VPN by default.