Before we begin, please note that 3.x is no longer supported. That goes with 3.x upgrades as well. If you run into issues, we recommend working with a third party provider to perform your upgrade.
In regards to what you're stating here, you would need to perform an upgrade, there is no way around it. While having a working 3.x installation is ideal to perform an upgrade, that is getting increasingly difficult due to the technology gap. With that said, you just need the database and attachments to move forward with the upgrade from 3.x to latest of 4. You would simply move those items over to your new server along with your conf_global.php file (don't forget to update for your new server) and then go to /admin/upgrade .
It is not possible. The upgrader handles a lot of different schema and data changes required. It isn't simply renaming.