Thank you both very much! So what I did now is this:

Also replaced the soil that it came in from the supermarket (which definitely was not ideal and quite slow draining) with some cactus stone soil stuff.

Should I just water it normally (and yes, definitely a bit less than before :D)? Or do anything special until it roots?

And I now have a lot of the lower leaves to try to propagate. I hope I did not pick off too much:

Will also try your bottle method for some of them. :)