Not sure what you mean by “directly,” there are going to be strong financial ties to any VPN provider one chooses, and as such, they are no better than running your own in a lot of circumstances.
Not if you pay with cash (such as Mullvad). Then it depends on what they can infer from your traffic.
If you have a server, allocated an IP, and someone sees that IP... that's a pretty direct tie.
If your crime is downloading Frozen, and the authorities can’t find any trace of the movie on the device the logs claim was used, then it would be shameful if the logs alone were enough to convict the person.