You have to look into each catalog file and determine if you want to continue using it. Removing images will cause you to lose your edits and user-supplied metadata, so I don't recommend you remove images.
Furthermore, I recommend you use one catalog for ALL of your photos, rather than a multitude of catalogs. You can combine the catalogs by opening one of them, then use the menu command File->Import From Another Catalog to combine.
Additionally, you can open the Adobe Creative Cloud application > go to apps panel > click on Open next to Lightroom CC2015 (if you have it installed) > that will open your Lightroom CC catalog.
Same goes for Photoshop CC2015 as well.
If you have Photoshop installed, it would say Open in front of Photoshop CC or else it would say Install.
dj_paige and others, Thanks for your advice to put everything that is
important into one catalog that is associated with the cloud. It's what I
need to do to end this confusion.
Before doing that, I'd like to install Lr 6 CC in my computer and then,
one by one, "import as catalog" those best images into Lr6 cc. Only at
that point do I, one by one, remove the several catalogs of Lr 5+ I have
lying about. Will this work? Will the website allow me to download
another copy of Lr 6 cc? On this laptop, I do have a "Adobe Lr 6-2.lrcat",
but I want my main catalog, the one going to the internet, be on my desktop
which doesn't have Lr 6 cc. Will this work?
Before doing that, I'd like to install Lr 6 CC in my computer and then, one by one, "import as catalog" those best images into Lr6 cc.
As a matter of philosophy, I don't recommend using Lightroom this way. Lightroom is the tool that lets you organize and search for ALL of your images, not just the best images. You want to import ALL of your images into one Lightroom catalog and then manage them, organize them and search for them in that single catalog. If you use Lightroom for some other purpose, like "best images" only, then you have created extra work for yourself, and the possibility of mistakes will creep in as time goes by, when you don't remember whether a particular image is in Lightroom or not.
Also, let's clear up some confusion here: you refer to LR 6 CC, but there is no such product. There is LR 6 and there is LR CC (actually LR CC 2015), and these are different, with different pricing and different features. LR CC 2015 allows you to synchronize your photos to the Creative Cloud so they can be viewed on mobile devices. If you want to share only the "best images" on your mobile devices, this is easily done from within the one catalog (with all of your photos) that I hope you will create.
Only at that point do I, one by one, remove the several catalogs of Lr 5+ I have lying about. Will this work?
If you are 100% sure you don't need the LR 5 catalogs any more, you can delete them. But please be 100% sure, not 90% sure or 99% sure, because once those catalog are deleted, any work that was in them would have to be re-done.
Will the website allow me to download another copy of Lr 6 cc?
You can download LR CC 2015 or LR 6 as many times as you want.
On this laptop, I do have a "Adobe Lr 6-2.lrcat", but I want my main catalog, the one going to the internet, be on my desktop which doesn't have Lr 6 cc. Will this work?
There is no easy way to synchronize the Lightroom catalog on a laptop and a desktop computer. The best solution is to put the catalog file on an external HD that can be plugged in to the desktop when you need it there, and plugged into the laptop when you need it there.