I thought I previously synced a desktop and a laptop but if I did (at this point, from what’s visible, I can’t tell for sure), the two computers are not synced (more history & bookmarks are on the laptop than on the desktop).
You first need to connect to an existing sync account to be able to manage that account.
Is sync installed and working on the desktop computer?
Both devices need to be online at the same time if you want to pair a device with the (3x4) 12-character code.
Sync needs to be set up and working on at least one (desktop) computer, so you can enter the 12-character code, provided by the device (computer) that you want to pair, in three fields to pair a device on the computer that has a working Sync setup.
Enter all details like the account name (email address) and password and the 26-character Sync key manually to connect to an existing sync account instead of using the 12-character code to pair a device if pairing isn't possible.
open "Tools > Options > Sync" or "Firefox > Set Up Sync"
click "Set Up Firefox Sync" to go to "Firefox Sync Setup"
click "I Have an Account" : Connect
click "I don't have the device with me" at the bottom to enter the account settings (email, password, sync key)
Check the Sync options when you have used Sync before and want to keep previously stored data on the Sync server.
click the "Sync Options" button at the bottom to select what to sync (Sync Options only affect the first sync)
Select this choice if you want to restore data from the Sync server:
"Replace all data on this device with my Sync data"
I don't want to lose either the data that's already been synced (on the server?), or the browsing history data on my desktop or laptop that exists since the sync. Is there another option besides 'Replace all data'?
Also, I know this sounds dumb, but I haven't been able to see what exists on the server. Can I do that & if so, how? Can I download it & if so, how?
If you want to merge your local data with whatever was already synced, choose "Merge" in Sync Options — it's the default.
There's no way to see what's on the server without connecting a Firefox profile. It's all encrypted. If you wish, you can create a new Firefox profile and connect that — once it finishes syncing, it'll have a full copy of everything on the server for you to look at (and you can even modify it and the changes will sync back up).