Are you doing your installing from installation files from Adobe or from an installation disc?
There is no guarantee that a project created in an earlier version of Premiere Elements will open in an later version. It should, but no guarantees.
Safety net approaches - finish projects created in version 10 in version 10 or work from a copy of the 10 project in opening
the project in the later version 13. And, above all do not overlook that the a project needs to trace back to its source media. So, consider using the Premiere Elements 10 Project Archiver Copy Project option to generate a Copied Folder that you can transfer to a USB Flash Drive for transfering the resulting Copied Folder to the second computer. If disc menus are involved in the Premiere Elements 10 project that you are archiving, further considerations are needed.
As for the Elements Organizer...I came across this Adobe Photoshop Elements Forum thread that has the flavor of what you
are about to do.
But, that seems to have been done all on one computer where catalog conversion would be my first choice.
But, yours is from computer to computer coupled with version to version. If you have a lot invested in the Elements Organizer 10 catalog,
I think that I would ask advice from the Barbara Brundage, MichelBParis, DJ Paige group in the Adobe Photoshop Elements Forum.
to minimize dangers of compromising a catalog and its tagging. I hesitate to offer how I would approach this because Elements Organizer 13 is new, and I have not explored it fully yet.
Thank you for the advise. I have P10 from Adobe download - which is another issue, will I have to disable P10 before downloading P13?
Thanks for the follow up.
You do not have to to deactivate and/or uninstall Premiere Elements 10 before downloading and installing Premiere Elements 13.
Just do not run the two of them on the same computer at the same time. The suggestion is to do the downloading and installing
with antivirus and firewall(s) disabled.
We will be watching for the outcome.