It all depends on your IPTV provider. But yes, it's working fine on both Fire TV and stick
Yep, as others have already stated, Fire Stick works just fine with Kodi & its ability to stream IPTV.
There's some tips and such to ensure you clear out unnecessary files from time to time, but other than that it will work just fine.
One thing to keep in mind, if you like to use community builds or anything like that, they can be incompatible or not optimized for use on the FireStick. It's a very capable piece of technology, but there's certainly a limit of how much it can handle. Always suggest to configure Kodi from scratch on one of these device's.
I am using cCloud and I watch some of the Indian channels and they work well on Fire Stick. It is more dependent on the source here and not your Fire Stick.