Put the catalog and the previews on the same external hard drive in their own folder. When backups of the catalog are done, make sure they are saved on a different hard drive from the external hard drive.
I had been thinking having the catalog on external hard drive would slow things down…guess that’s not an issue?
And do you happen to know: will the catalog be readable/written OK w/ me having LR3 on desktop and LR5 on laptop?
Running the Catalogs and previews from an external drive may slow things down, it depends very much on the type of drive and its connection to your computers.
However, your biggest issue will be running different versions of LR on the two machines. LR5 will want to upgrade the LR3 catalog and LR3 will not be able to use the LR5 catalog. You need the same version of the software on both machines I'm afraid.
Lightroom 3 and Lightroom 5 use different catalog formats, and are not compatible with each other. You need to be running Lightroom 5 on both computers. Your license entitles you to two installations for your personal use. The only reason not to upgrade the Lightroom 3 is if the operating system won't support Lightroom 5. Having the catalog on the external drive could have a slight impact on performance. I have done it a few times. Most notably was when I was working on wedding photos from my daughter's wedding. I didn't notice any performance issues at all. If both computers support USB 3 then I would make sure the external hard drive does as well.
It was in fact an OS issue: I've been a Capture NX user for years, and have clung to Windows XP because of it, but now that I've gotten a taste of Lightroom, and liked it, I guess it's time to bite the bullet, move over to Lightroom 5 everywhere (have LR3 on desktop), and get Windows 8. The time has come!