The PSE software is available to be installed and run on a computer (no other types of devices such as tablets).
You get the installer file either by download (only solution from Adobe directly) or on a DVD (from big Web resellers).
The license is for a single user.
The installer, download or DVD can install the software itself on any number of computers.
To be able to run a given computer, you have to 'activate' the link with the licensing server from the menu Help >> sign in. You are allowed to 'activate' two computers, typically a desktop and a laptop at the same time, but you should not use them at the same time (single user license)
If you have already activated two computers and want to activate a third one, you have to deactivate (menu Help >> Sign out) one of the already activated computers.
So, the answer is that you can have your four computers with the program installed with your downloaded or DVD installer.
You can have two of those activated at the same time.
You can switch the activation for two computers in the same location.
But you are not allowed to use or let other users use two activated computers at the same time.
Sharing of your media and catalogs:
You can backup and restore your catalogs and media from the organizer, which allows keeping copies on various computers as well as allowing to migrate your data and organization from one computer to a new one.
In your case (just like me), I would store the catalog AND media files on an external drive which I can plug in to any activated computer without having to do anything else.
To be precise, there is no 'syncing' in the process. Your catalog and photo/video files are always updated and available to the computer to which the drive is presently plugged in.
'Syncing' would mean that a copy of the catalog and files would be updated when you plug in the external drive. If you want to keep copy of the catalog and the photos updated in two computers, the solution offered by the organizer is to backup the catalog of the first computer to the external drive and to restore it on the second computer. That takes time, but that's a valid choice if you change computers only a couple of times in a year.
Real 'syncing' with external tools like Microsoft Synctoy is a different matter. Let's say you start with a backup and restore from the main computer to the external drive. The external syncing tool is apt to update the photo/video files as well as the catalog folder on the second computer. That works well and much faster than a backup/restore, but there is a trap: the 'synced' catalog folder does not point to the files in the new computer, it still points on those on the first computer. All files are 'disconnected' and reconnecting a whole library is hardly possible in the organizer (it would be easy with Lightroom). You would have to fiddle into the database with an sql utility.
The 'syncing' problem would be the same with a Cloud solution like Dropbox or OneDrive instead of an external drive. No problem to sync the photo files, same problem with the catalog folder.