To answer your question: Yes. You can add gelatin to a carbonated, kegged beer. I've done it before, actually quite a bit. One thing to be cautious of, adding gelatin to a carbonated keg will causing the beer to foam rapidly. If you do it, I suggest pouring the gelatin in with one hand, and having the lid to the keg in the other hand, ready to quickly seal off the keg before it foams over.
It's important to note though, and I welcome others saying otherwise as I'd like to hear their process, but from my experience, when I forget to add kettle finings, gelatin is not sufficient enough to clear the beer by itself. If I let the keg sit several weeks after I fine it, it will pour clear after several pints, but by itself, gelatin, doesn't seem to do as well as when it's used in conjunction to kettle finings like Irish Moss.
That said, it does look remarkably better than no fining at all. I'd recommend making sure the entire keg, top to bottom is near freezing, and perhaps maybe using a bit more gelatin than usual.
Follow-up on your additional questions:
I add approximately half a gelatin packet to a mug filled approximately two-thirds full of warm water, let it rest for a moment to ensure all the gelatin is submerged (if you don't you risk clumps). I then microwave it for about 1 minute to get it up between 160-180°F to kill off any bacteria, then add it directly to the keg. Some people have come back debunking the theory of why boiling gelatin in the water is a bad thing. My process works for me, it's easy and consistent, so I see no need to change it.
Also, yes. You need to ensure your beer is as close to freezing as possible to ensure that chill haze has come into effect, the primary factor that will turn a cold beer cloudy. If you're willing to put in the effort to make sure it's a consistent temperature across the entire keg (I've seen a lot of homemade keggerators where the temperature can be off by quite a bit, 10-15° worth, between the bottom and the top of the keggerator), you can add your gelatin, purge the keg with CO2, then give it a shake to make sure the temperature is even throughout. Don't do this before you add the gelatin though, for the same reasons you don't want to shake a beer can or a soda bottle before opening it.