Hello! Unfortunately, there isn't a way to do manage multiple versions of the driver within the same project. There is kind of a workaround where you can separate with modules in Java, but I've been told that is very difficult.
Yes, the driver is typically related to the version of Neo4j itself. However, many of the driver versions are backward-compatible with previous versions of the database. You would need to look up whether the driver version is compatible with your version(s) of Neo4j to verify.
As far as upgrading versions and a method not being used, there should be documentation pointing you to the replacement for a deprecated method, and some of our error messages or deprecation warnings also do that. It shouldn't be too much trouble, but please do let us know if you go that route and have issues!