One more distro under consideration: A distro for "home server/office/family". That is: it should share some disks and a printer to other machines in the LAN, and it should be easy for not-so-computer-literate to use for web surfing, photos, social media etc.

The HW id Gigabyte B75M-D3H MoBo with i3-3220 processor (ivy bridge, HD graphics 2500) 4GB RAM.

At least CUPS should be at least version 1.6 (1.5, still has bugs that makes it not working with my printer).

SW doesn't need to be the cutting edge, but not very old either. Smaller children may want to play some java/flash games.

Oops, I forgot: No smart phone OSs.

Any distro.
Any distro that offers you the ability to install (or already comes with) a DE that is considered to be user-friendly, usually KDE, Mate, Cinnamon, Unity and Gnome. The last two might be what you consider to be "smart phone".

Regarding CUPS, if you are thinking about a distro you can use Distrowatch to determine which CUPS version it comes with.

The "Not-So-Computer-Literate" can mean anything, but as long as a distribution is well-documented, anyone with the ability to at least read English should be able to get the idea of what to do.

To me, Slackware is an ideal choice for you. It's possibly one of the most well documented distributions and even has the installation guide on the disk, several online FAQs, hundreds to thousands of users who know the system, and a book you can read up on as well. It may seem a bit non-user friendly at first, but it's really one of the simplest and maybe easiest distributions to use.

To be honest with you, you might learn more and be more comfortable with a seemingly difficult distribution than one that's more read-to-use. Having hands-on to teach you things, really will help later on.

Seeing that the OP is searching for a distribution used by family members and and maintained by the OP it is actually irrelevant how well documented the distribution is, as long as the OP is able to maintain it. My recommendation would simply be to use the distro the OP is most comfortable with, possibly in a LTS version, if available.

Slackware, by the way, is still using Cups 1.5.4, even in -current, so it is no option here, unless the OP plans to upgrade Cups (and possibly packages dependent on it) himself.

As much as I think Slackware is the bee's knees and the cat's meow, I'd be inclined to recommend Mint, as it comes with Flash and with every codec your family might every want to use, and Mint 17 will be supported until April 2019.

TobiSGD got the idea.
I'm currently running Mint 17 / Cinnamon, but for some reason my printer doesn't work with it (Canon Pixma MP 140). The printer worked with Xubuntu Live DVD, and after that with Mint Live DVD, but when I installed Mint, the printer stopped working again.

What I really meant with "smart phone OS" is Ubuntu/Unity.
I can't stand it that I can't browse the programs. If I need, say, a lightweight web browser, what name should I search for?

Also, the kids can't read English too well, because they are mostly pre-school aged Finnish kids.
Picture (icon) is essential.

Are there good candidates in the "rpm-world"?
(I'm somewhat familiar with the "deb-world" already.)

Even though I am a Debian user through and through your question regarding an RPM distro would indicate Fedora may be worthwhile looking at.

I would consider Fedora, at least until they come further with their Fedora.Next plans, to be unsuitable for a machine that also acts as a server, both due to its short life cycle and its frequent changes of software versions.
Possibly the just released CentOS 7 would be a better fit.

TobiSGD, have you been where I'm now, or how do you spot essentials that well?

My wife HATES technology and gets irritated VERY quickly if her computer doesn't do exactly what she wants it to do. She has no patience at all and just wants it to work. I finally set her up with MX-14 and she has been very pleased. It's stable, works well, and has all the features she needs. She is a basic user and surfs the web, occasionally works with a spreadsheet or document, uploads & downloads pictures of the grand kids, copies pictures off her phone, listens to her music library, and uses Skype to talk to her family in the Philippines. It comes with the Xfce DE but I installed LXDE and removed all the menu stuff that she will never use to give her a simple and clean interface. The support forum has some very smart folks and I've seen problems solved very quickly even if it involves old hardware or unique needs.

It's based on Debian Stable but the authors bring in stuff from backports so the software is "fresher"...

http://www.mepiscommunity.org/mx

I just like to evaluate the tools I can use (and an OS is nothing more than that) for specific workloads/environments, before I decide which one to use. Fedora is a good distribution if you want to live on the bleeding edge and your main focus is development, especially when it comes to Red Hat technologies. But those feautures make it in my opinion unsuitable for servers that have to be supported for a long time without dealing with frequent updates and where stability is key.

Sounds like worth ckecking up.

---------- Post added 07-08-14 at 04:08 PM ----------

Am I re-inventing a wheel here... ;-D

This is a very true statement and matches the way I use a computer perfectly.

In general, I want my OS to stay out of the way and be transparent for the most part. I am no longer a programmer or developer - I'm just a user that wants to efficiently run the applications I want to run. It took a while to find an OS my wife would be happy with as it had to be stable, reliable, have access to the programs she uses, and be absolutely no hassle - and MX-14 does this well...

Well, I'm still a programmer, but I still want to do my programming instead of wrestling with the environment.

I like OpenSUSE 13.1

I use that as my everyday OS and ScientificLinux 6 for the programs that NEED that

suse is in the RPM redhat family ( sort of ?? )
they use rpms but are VERY VERY gui centric and dose things a bit different

most of the time it is EASIER to use the gui over the terminal