So, my use of kdenlive just came to a grinding halt: I can neither work on my old project, nor work on any newly created project, as I can no longer edit effects and transitions. Before I take my loss and look for an alternative, let me try to get this straight.

Page https://kdenlive.org/download-ubuntu suggests the recommended version for Ubuntu 15.04 is Kdenlive 15.04.3, so I need an downgrade, as I use 15.12.0.
A number of bug reports I submitted were answered with the claim that the bug had already been fixed in version 15.12.1?

What is currently the most stable version of Kdenlive, and how do I install/upgrade it?

15.12.1 is stable, also 15.12.2 ... which is recent stable yet most distribution repositories don't have it. If sunab's repo shouldn't work then compiling from source is a good alternative. The kdenlive website has a link to the Wiki page that describes compiling Kdenlive on KDE Frameworks 5.

Your best bet is probably to get Ubuntu 15.10. The 15.04 reached EOL (end of life) on February 5th. That means it's not supported any longer, no security updates. I had Ubuntu Studio 15.10, until recently, and adding the Sunab repositories, was enough to get Kdenlive 15.12.1, with all icons in place. Ubuntu Studio 15.10 has Kdenlive 15.08 preinstalled. In 15.08 there was a missing icons problem.

Now I have Ubuntu Studio 16.04 Beta, with Sunab repositories, and Kdenlive 15.12.1. To be clear, I do NOT recommend running a Beta version as your main OS.