Likely some type of anemone. Keep an eye on it and make sure it doesn't start to proliferate. Some can be trouble. It looks like it's irritating the polyps around it. I'd hate to have those all over my tank as happens with some problem anemones.
Yeah, frag means just that. As it gets bigger you'll be able to break branches off and trade then in at your fish store, or sell them to people. Helps make a little dent in the cost of this hobby and decreases the number of corals taken from the wild.
It's a Stomatella and is harmless...in fact they're kind of nice. I had some in my tank for a time when it was newer, but they all eventually died off.
I have my green branching frogspawn about a foot and a half under 2 96 watt PCs. It's thriving there...had it for 2.5 years and it's developed at least 20 branches from the original 3 it had. It's my favorite coral. I've tried to feed it a few times but it rejected the food each time so I've...