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You can only activate Lightroom on two computers at any one time although you can install on all three.
The catalog is constantly being written to so cloud-sync could affect performance. Catalog size may also eat up storage unless you have paid storage. But in theory it could work provided you allow time for everything to sync before doing new work in the catalog. Mike Hoffman uses Dropbox Sync, but for the smart previews rather than the catalog. See if his tutorial at Tip Squirrel helps.
I had read Mike's article but I thought that this solution would only allow me to add new photos at only one machine whereas the 'Sierra Sync' solutions allows at multiple machines. I only use one computer at once, one is an older iMac used 99.9% as a target display monitor, the other two I switch between if I'm at work or home.
I have run into some catalog issues today. The photos were in the sync'd photo folder but not in the sync'd catalog, 'Sync Folder' (sp) solved the problem but of course I lost my edits. Not sure how this happened.
No more thoughts on this topic? :-)