The Paralog installer will only dowload and install the latest Java Runtime Environment if no suitable JRE can be found on your system.
This installation will be local to Paralog. It will not available to other applications.
Download and install the latest version on the new machine.
Register with the license key from the original registration email send to you.
Copy your data directory to the new machine (File/Open on the main menu will show you the location).
Use File/Open from the main menu to open your log file in the copied directory.