I'm not Vincent, but I'll claim proximity counts for this one :) Also, I've played multiple characters since I started playing, back 30-odd years ago.
1) Have you or the groups you've taken part in done it, how often, and with what systems - things like that? It was a completely alien concept to me before AW, so I'd love to hear more about it. What's good about it when it works, and what can make it go wrong?
I'd say we wind up with some shade of multiple character roughly 50% of the time, regardless of system. Sometimes it's just as simple as a player 'doing the voice for' a minor NPC related to the player's PC.
Example 1: We're playing Dogs in the Vineyard, and your Dog is talking to the NPC cousin of my Dog, in this town we just rode into. The GM might be running the Steward in the scene, and a couple other NPCs, and then your Dog might ask the cousin "So where can we rest up before tackling this infestation of rats you have here?" and I might 'do the voice for' that NPC -> "There's a good dry barn out back of Uncle Hank's; I reckon he'd welcome you there." I'd probably glance at the GM first to be sure s/he wasn't planning anything, then next time we saw that NPC, I might 'do the voice for' the cousin again. It's not a fully statted up PC, but it's one way it happens. It makes for a richer, more diverse set of NPCs, since it's not just one brain running them. In a game like Dogs, with a strong GM, the GM gets veto power over any NPC of course, no matter how much I've been 'doing the voice' for it.
Example 2: Sometimes we've played games where we're going into a big event, like a town meeting or a wizard's council, and we go through and decide who's going to run which of the masses of NPCs. We still recognize that there are PCs and NPCs, but if it's you and me and Mitch (hi Mitch!), and we've got twenty NPCs at this big event, it's masses easier if we just divvy them up, talk a bit about their motives, and then just play them. This lets the GM not worry about keeping everything in her head, and instead she can think about story arc and fronts and cool details. When the big event is over, our PCs go home and the GM might make some notes about stuff that happened, in case she wants to bring any of those NPCs back, but it's not expected that we'll see them often.
Example 3: The other is when you do stat up two (or more) characters. How that works is I make two characters (or a second to bring in, as with AW), and play them both. It's a lot like cutting back and forth between plot-lines in a movie or TV episode or book - sometimes one character gets more screen time than the other, they are not always in the same scene together, and it gives me two points of perspective on the action.
That last is a HUGE bonus, and leads straight to the next part of your question: What's good about it when it works, and what can make it go wrong?
What's good about it is the ability to be two places at once! I'm currently playing Mox and Keeler in our AW game. Mox is the Angel, Keeler's the Gunlugger. Having two PCs means I'm twice as likely to be in a scene with another PC. I can have Mox be hanging with Brace, so I can take part in whatever's doing at home *and at the same time* I can have Keeler on a shopping running to Harridan's! I could be in simultaneous scenes with Rose and Amanueal (r.i.p), even when they were on opposite sides of the world.
The other major good thing is the ability to approach the setting from different angles. I can play Damvild the Bjornar, who's fairly pragmatic and family-oriented, and Oldoyn the flighty and weird Criamon, *and* the disarmingly friendly and slightly disturbing Jerbiton mage Ardesco. Playing multiple characters gives me more ways to engage in the over-all story, and more options in a given scene as to how I want to take part. Note: sometimes it's great to just play one character!
Brass tacks: If you're going to try it, make sure the two PCs are whole and unique characters, that you feel you can inhabit and play with solidly on their own. They need to be distinct, so your fellow players catch which is 'on' when you talk, especially if both PCs are in the same scene. I think we all unconsciously sit and speak a bit differently when we are in character; figuring out how your two PCs differ in those regards is a really helpful way to clue your fellow players in.
Things I noticed I was doing when I was playing Mox:
Mox is a bigger, older guy, who speaks with a tiny hint of uncertainty at the back of his voice, which is deeper and softer, less clipped, than Keeler's. Also, Mox tends to sit back in his chair a bit more, and he fidgets with his fingers a bit when he's worrying something in his mind.
Things I noticed I was doing when I was playing Keeler:
Keeler speaks clearly and with authority, but she's not in it for the love the way Mox is. If it's not her problem, it's not her problem. She doesn't fidget, she sits solidly but leans forward on the balls of her feet, ready to move.
Also, even the most quick-on-the-pick-up folks are going to mistake one PC for another sometimes - don't sweat it.
If it's not working, it will be because one character is just being sidelined, not getting enough screen time to flesh out into a full PC. This is fine if they are sort of taking turns in the spot-light, but not fine if you are just not playing that character. If you're not interested in them, in that angle, it's probably better to either let them fade into NPC status. If there's an MC or GM, they might be forgetting about that PC, especially if they are new to players running multiple PCs. In that case, you need to step it up and get your second PC involved in the storyline. Walk on-scene, say what that PC is doing ("Meanwhile, I'm over at the AutoMart stocking up on rusty screwdrivers"), get into a conversation with another player's PC. Playing multiples is not for everyone, or for every game.
2) Would you say this works in any of your other games? Could a player go into Dogs with two characters, or IAWA with multiples? What do you think the pros and cons of such an approach are? Would any rules need to be tweaked?
Some games are way more supportive than others - it's easy to play multiple PCs in In a Wicked Age, frex, but not so much with My Life With Master. 1001 Nights assumes the ability, and also teaches how to do it, to some extent. Dogs would be pretty challenging, I think, but not impossible. I wouldn't want to lead out with that as my first multiple PC game, anyway. I'll let Vincent speak to that more directly.
3) And lastly, are there any games out there that you think are really, really good when a player plays multiple characters, regardless of whether or not the rules encourage it?
We played Ars Magica for years - probably almost 10 years, if you combine the various games - with each player running multiple PCs. I can't imagine playing it any other way, and it really was the first to suggest 'hey, this is a cool option', alongside troupe-style 'GM-less' play. I think Primetime Adventures could work, although spot-lighting episodes would be a bit trickier. No real reasons I can see why Sorcerer and Misspent Youth couldn't work with multiple PCs. As far as rules encouraging it. I'm a bit hazy.