Is testifying on a car bill of sale, in which purchase and payment of the price were claimed, considered a false testimony, or is it permissible if the original owner voluntarily relinquished its price? And is it permissible to rectify this by paying a symbolic amount?

The default is that lying, forgery, and bearing false witness are forbidden. However, if there is a fear of rights being lost and such an action becomes the only way to preserve them, then there is no harm in it. Therefore, if the gift could not be documented except through the method mentioned, then there is no blame upon you for what you did. As for paying a symbolic amount later, it is of no benefit, because the testimony was about a past event.