Basic advice: Consider how you'd react at work, or some club you belong to. Meaning, look at other arenas where you could really hose the group dynamics. That takes the geek factor out of it, to an extent. It also removes it from your home (or someone else's), which really isn't relevant, but could color things.
fusangite said:
1. A number of people have suggested including this woman in a group social activity in the near future; it is a direction I was thinking of going. But then it occurred to me: should this activity include (a) no other people from the game, (b) one or two other people from the game, (c) most/all of the gaming group?
Depends on your group dynamics. My gaming group is also my circle of close friends, and that's been the general rule. We've had non-friend (i.e. casual acquaintances, and game-only contacts) join. But there's about an 80% chance that getting into the game meant you were on the barbeque, movie, go to Ren Fest, etc. list -- or that getting on that list gives about an 80% chance you'll be invited to join the game. If you leave the game, you'll probably slowly fade off the radar and if you stop calling us casually, you'll probably not be invited to the next campaign.
If that's your situation, then the most natural thing would be to just start including her in group outings. After a couple of outings, you can call her up and say, "Hey, why don't you and I go see a movie/play at the arcade/check out this new restaraunt/whatever."
If your game group has different dynamics, then revise the plan accordingly. If you never hang out with other members of your game on a casual basis, then getting people together just so you can hang out with this gal will be unproductive at best, ackward at worst. Just contact her (phone, email, after the game, whatever works) and ask her to do something. This'll be the most obvious date-feel.
You could also invite her to do something with whatever friends you normally hang out with. That's less datey, and more, "Hey, I think I'd enjoy hanging out with you." Depending on your goals, the latter could be an even better thing than the former, at least as an ice-breaker. There's nothing wrong with this. Your goal is to spend time with her and connect personally, not to isolate her from society.
In a similar vein, if your group is a mix of friends and game-buddies, then you can ask her to join some activity that your friends are planning (movie, dinner out, paintball, whatever). Don't invite anyone from the group who isn't part of your posse besides this gal. That keeps it, quite clearly, not a game thing.
Note: I'm assuming that you live like I did in my 20s (regardless of age -- I'm comparing no kids, lots of free time, etc.). That is to say that you often do group social activities like movies, you don't have kids running around, and you have a sort of "posse" that you generally hang out with casually. If that's not the case, then I suggest you ask her out in a straightforward, "Why don't you and I go do X," fashion.
I'm also assuming you're looking for a long-term possibility, at least more at that end than a cuddlebuddy. If it's the latter, then my advice is to forget it.
2. A number of people have suggested just asking her out by telephone in the near future; it is also a direction I was thinking of going. But my follow-up question is: how should I deal with any potential social awkwardness if the next game falls between me asking her out and the time that we end up meeting?
If your buddy Bill, who games with you, asks you if you want to catch Sin City this weekend, how would you react? If the game fell between when he called you up and when you were planning to see the movie, you probably wouldn't think anything of it. You wouldn't say, "So, are we still on?" because you'd know that you already agreed on it. You wouldn't stare at him all night, but you wouldn't avoid all eye contact, either.
If it comes up, it's not a taboo subject, but "Bill and I are going this Friday" wouldn't be your first words unless you were planning to invite everyone else. More likely, you'd say, "Yeah, I'm really looking forward to seeing it." Everyone who needs to know the rest (i.e. the gal will understand you're including time with her) will get the point.
Don't avoid her company, but don't follow her around like a puppy.
3. How forthcoming should I be about this project with my other players?
IMO, this comes back to whether the rest of the players are part of your "posse". If they are, then talk normally with them -- like you would with any potential date. If they aren't, then they don't need to know details about your personal life. That said, if there's one or two that you might have a frank discussion with, and they might be good sources of advice/insight here (if you're lucky, this includes whoever brought her to the table), then ask them very openly and honestly.
There are a lot of variables. This is really where the "treat it like work" idea comes in handy. Would you ask just any random coworker about dating a new coworker? Probably not. But, if you had a fairly close relationship (occasion beer or some such) relationship, then you might.
4. I have to continue producing lots of background material for the player because her character's origin is different from that of the other PCs. Are there ways that I can avoid or limit this ongoing activity being recontextualized in light of me asking her out?
I would totally and completely avoid using anything game-related as an excuse to get together. Never mistake spending time working with her on her character as an implied date. If you want to use it as a segway of some sort (which might be legit), say, "Hey, I think it'd be nice to do something that didn't involve the game. How would you like to grab some coffee/ice cream/club baby seals after the next session?" Modify based on circumstance or desire.
Caveat to free advice: You get what you pay for. I've been married for ten years, so I'm rather removed from the dating circles. But I'm very comfortable with talking to women I want to date (which is a field of one).