> Is the hardware of PinePhone considered stable?
There has been some hardware improvements between the braveheart edition and KDE Community edition. For example the convergence functionality was added. Afaik the Manjaro and KDE edition will be using the same hardware.
> How is the development on such platform, the same a on Desktop GNU/Linux? Do we have a permission system similar to Android? Can we launch easily background application and awake the phone when needed (for notifications)? Is there a documentation on the packaging system?
The development for Plasma Mobile is similar to the one on the desktop. You can ever develop completely on the desktop or ssh in the phone and compile there. Packaging system is the same as on the desktop and depends on the distro. Flatpak also works as cross distribution solution ;)
KClock uses Powerdevil (the kde power management deamon) to wake up from sleep. There is also the KDED infrastructure: https://api.kde.org/frameworks/kded/html/index.html.
> How long can the battery last at the moment (with mostly idle phone)?
Good question. This varies a lot by the software used and the distributions. At the beginning it was less than 2 hours but with a lot of improvements in the kernel, there was reports varying from 12 to 16 hours idle time. This will probably improve further with more improvements in the kernel and in the hardware stack.
I hope this answers all your questions.