Have you seen this coral before?

yes, that is a purple fungia

Whatever it is, it is very recessed. Fungia would be my first guess, but it could be something related like a scolymia. Have to see a better pic, one with it inflated so we can see the tentacles.

It's a fungiid. Maybe something like Halomitra.

Rare, no. Uncommon, yes.

Fragged it? Doesn't look like a frag. I thought things like fungias the only way to frag them is to cut them in half/quarters making sure to get a piece of the mouth on each frag. The normal way they spread is they just pop out babies, mini versions of themselves. Actually seen that more often than I have seen people fragging them since they are one of the more difficult corals to frag successfully.

I will have to agree then with Amphiprion. It is a Halomitra, which is similar to a fungia, but not the same. Hopefully Amphiprion can chime in and say if the manner it was fragged was ok.

I can say I have never seen one of those before, and even google comes up with very limited information on them, so I would guess it is semi-rare at least.

Edit: From this picture it does appear to have multiple mouths..