Ok, with the narrowing qualifications, here's a few observations.
(1) If you try to date her and things go badly (and there are degrees within "badly"), there is no way that is *not* going to affect what happens at the game table. That is part of the risk. Best case senario is that interaction is a little stilted for a while but finds a way to be comfortable in time. And that's best case.
(2) All your other questions (about calling and hanging out and whether or not to tell you friends) all boil down to piece of advice. Treat her like a girl you might be interested in that you *happen* to meet in a gaming environment. If you don't make the fact that she's in your gaming group a focal point for your decision making when interacting with her and just treat her like a girl you like, I think you'll get a better response.
So, to extrapolate that piece of advice into your 4 points of interest....
Are you planning a social event or planning on asking her to someone else's social event? If you are planning it, then you would probably ask your other friends (some which may be in your gaming group) too. If you are taking to a friend's party, then take people who you would typically take to such an event (regardless of the gaming group status).
If you call her and you guys are going to go out after your next gaming session, well, then you go out after your next gaming session. It's only going to be as akward as the two of you make it. If you don't bring any weirdness to the table, that's half the battle won.
Should you tell other people in your gaming group? Only if any of them are close enough to you that you would tell them in the first place (again, taking away the gaming group connection from how you interact with the girl and make decisions about how to approach the girl).
And as much as it stinks, you cannot change the way she is going to react/reinterpret past actions when she finds out that you like her. So, the best relationship advice I can ever give (and this is golden so all you singles listen up)... Never do anything you'll regret and never regret anything you do. That advice is only as valuable as you know yourself. In addition, you also shouldn't do anything to get the girl that you would not want others to know about. And if you can't be honest to her about it, that is probably a good indication not to do it either.
Best of luck and if it doesn't work out, I have a cute cousin whose just finished medical school, if you are into former Chinese beauty contestants that are too smart for their own good.
-Suzi