Bilder auf ext.Festplatte von 2 PC organisieren und bearbeiten

Sorry to hear about your experience with Covid.

I am a little confused about what you mean by the international drive.  Is that an external drive or your internal C drive?  In any event, it seems that you have all of your photos and the PSE 18 catalog on this drive.  If that is the case, you should do the following:

• make a full backup of that catalog, using PSE 18 on your laptop.  (Perhaps you have already made that.) 
• Next, use PSE 18 on the laptop to restore the catalog (and all of the photos) to the new external hard drive which will be shared between the laptop and desktop.  I recommend that you create a new folder named, for example, My Photos on the new drive.  You will Browse to and select this folder as the New Location for the restored catalog:

• The Restore process will also restore your catalog database folder (named My Catalog unless you have renamed it) to the folder you have chosen.  In my example, this will be the My Photos folder.   I strongly recommend that you move the folder to a different location on your shared external hard drive.  For example, you could create a new folder on the drive called Elements Catalogs.  You can cut and paste the My Catalog sub-folder to that new main folder, using Windows File Explorer.
• Now, with the external drive connected to your Desktop Computer, use Windows File Explorer to go to the My Catalog folder on the shared drive, and double-click on the file named catalog.pse16db.  The catalog with all of your photos should now open in the Organizer.

If you need more details about any of this, take a look at this Adobe Helpx article.  I have also recently helped another user with the same question.  You may find that discussion helpful in answering your questions.

You can use the files on your external drive for both your laptop and desktop.  But, you will have to use different catalogs because you are using different versions of the program on each.  This may cause some issues because if you tag a file, using Elements 14, it will not appear automatically when you use Elements 18 (with a different catalog).  There may be some workarounds for keyword tags, but I don't think it will be possible for People, Places or Events tags.

The better solution would be to use Elements 18 on both machines.  You are entitled to install the program on 2 computers as long as you don't use both at the same time.  That way, you could move the catalog folder to your external drive and use that catalog on both computers.  Any changes you make on one computer will then be seen by the other computer (except for Auto Creations which will be created on each computer separately, I believe).