I think it should be dishwasher safe, it is completely made of steel (but I have to admit that I have never tried it, because rinsing with water has been enough for me until now).

I don't think so. I just rinse them with warm water and that's enough so far.

Yes, it is! And why not, it's made entirely of steel. However, according to the manufacturer, the lacquered models may scratch the lacquer over time in the dishwasher (but this does not affect the function of the bottle). I usually rinse my bottles quickly by hand and occasionally put them in the dishwasher for a thorough cleaning - so far this has not harmed them (and mine are also lacquered).