Real simple question, can I socket and unsocket a monster gem in and out of different equips freely?
Nah. It will vanish. The gem, that is.
According to few threads I’ve read you can’t level them and unsocketting gems will decrease their level by one. And if it goes zero, the gem goes poof.
I’m referring to Monster Gems, not common gems that are green,red and whatnot.
@Seiryuu oh well, thats unfortunate 
I know, but as far as I know, they work with the same socketing system and rules as regular gems. With the difference that you can’t enchant them, which means they will break when you unsocket them.
Sorry but I can’t confirm this myself because I’ve been saving all the monster gems I have. It’s what I’ve read on these forums. I don’t want to be the one who does extensive trial and error testing on them 
I dont think thats the end of the monster gems tho. The whole system looks unfinished, not well thought out, renderIng a useful item and super rare drop utterly useless.
The system warns you when you’re about to unsocked a gem, actually.