Does it works with sumsong ultra 23?
Well, it does say this in the product features:
“Wireless Charger Modes: Providing 10W High-Speed Charging For Samsung Galaxy”
So there’s a good chance it does.
Does fast charging hurt the battery over time? I know nothing about it so this might be a stupid question.
Fast charging means it’s designed to handle higher current to charge faster
23 ultra can support up to 45 watts charging- 10 watts isn’t really fast charging for newer devices 
I didn’t write it… just echoing what it says on the page.
Does not appear to include the AC adapter.
In general, heat kills batteries. So I theory this could kill your battery faster, if it heats up your battery. But it will also depend on your phone. If your phone is designed to handle higher watts, it should be ok.
Also, as others have pointed out, this is now on the slow side of “fast” charging
So would prolly be ideal for a nightstand, where something sits on it for an extended time.
I have one of these. Exact same thing, different brand name. Works fine with my Galaxy S23 Ultra.
Yeah, 10w isn’t really fast charging for any device. For example, my pixel 6 pro takes about 6 hours to charge to 100% on a 10w wireless charger. So it’s really only good for overnight use.
I grabbed one of these for use as a nightstand/overnight charger.
Unfortunately, the LED on the base is very bright and always on! Bright blue when no device is on the stand, bright green when charging, and (worst of all) bright flashing green when fully charged. And since this is an overnight charger, that means it starts blinking bright green at 2 am when the phone is fully charged.
The price was good and it works acceptably; I’ll likely cover the LED with a sticker.