Thank you for the answers so far. That helps a lot. I have already noticed that the apps are quite fiddly, as I already have a few other Hues.
Now the question remains how it looks after a power cut (via a switch on the power strip). The previous Living Colors kept the last setting. The Hue White switches to warm white approx. 80% after a power interruption, no matter whether it was switched on or off before.
The Lightstrip plus does not keep the last colour either, it also changes to warm white, but only if it was on before.

yes it does. the colour is stored in the profile on the mobile phone. but beware. the app is rubbish. complicated and fiddly. buy an alternative app.

Yes, the colour is always correct when I switch on. I have not yet had a power failure.