What is the recommended ROS distribution to use in 2019?

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Paragraph "5. Which distribution to use" at the distribution page has been last updated almost 2 years ago so I am wondering which ROS distribution is the preferred one. I know Melodic Morenia is the last ROS distribution [..]

and it's even an LTS (Long Term Support) which means it gets 5 years of support and development.

Section 5 does indeed look like it could use an update, but section 2 List of Distributions does state that Melodic is the Recommended ROS release as of this date.

I tried installing that on Ubuntu 18.04 but I couldn't do sudo apt install ros-melodic-desktop-full because I have to use a shadowrepository. This made me doubt the stability of ROS distribution Melodic Morenia.

I'm not entirely sure I understand what you're saying here.

What does "I have to use a shadownrepository" mean? And why do you (believe) you have to use it?

I will need to use ROS nodes other people have made last year in the project I'll be doing, those ROS nodes have been made on Ubuntu 16.04 with ROS distribution Kinetic Kame. Can I use those nodes in Melodic Morenia without too much hassle?

That will depend on how those nodes (packages) have been written, and how they are distributed / deployed.

Are they all packages for which you have the sources available? If so, there is a high chance that building them in a Melodic workspace will work. There might be a couple of things you may have to fix or update, but how much is going to be difficult to estimate as it would entirely depend on what these nodes actually do and what their dependencies are.

If you only have binaries (ie: compiled workspaces or .debs) then that will most likely not work (reliably).

Edit:

In the installation tutorial for Ubuntu it states that the recommended install command is sudo apt install ros-melodic-desktop-full which doesn't work for me, it can't find all packages. It says that I have to use shadowpackages if that's the case

Please copy-paste the exact message that you get from apt into your original question.

Using the shadow repository should not be done unless necessary, and from your description I doubt it is.