I think Allen is a genius albeit somewhat inconsistent. But I've found enjoyment even in his fluff movies (Scoop, The Curse of the Jade Scorpion, Everyone Says I Love You etc). That said in response to the questions in the original post:
What's your favourite Woody film? Or which do you think are his best?
It's hard.... I'll say Crimes and Misdemeanors and Match Point, which I see as sort of bookends that deal with similar issues and questions. I think Match Point got more into the idea of randomness within that question of getting caught. I remember when I was in high school we were discussing Crimes and Misdemeanors in the context of Machiavelli. And later that same year the film came up in a discussion about Crime and Punishment. I don't think Match Point had been made then. In both of those films people you wouldn't normally be afraid of (they're not criminals, they have no history of violence, they're respected, productive members of society) do terrible things. They commit their crimes as a result of a threat to something they value and because of their privileged position as a "good" person, they get away with their crimes.
*highlight to see there may be spoilers*
I guess that idea interests me. We really feel comfortable or uncomfortable with people based on very superficial factors. Does s/he have a job? a family? Then s/he must be "good". Does s/he have a criminal record? a substance abuse problem? S/he must be "bad". The films illustrate the obvious fallacy in that thinking (you never know what happens behind closed doors). But more than that they ask if their characters are evil on the basis of a single action.
What do you love about his movies?
I like the neurosis that he often depicts. And the themes of alienation and infatuation that I often see in his work. I like that he can get some of the best living actors in a film doing their best work. And I like that it can make me think and make laugh- sometimes even at the same time!
What don't you like?
I often have a problem with his female characters. I tend to get the sense that they're one dimensional and serve as a plot point or a step along the journey of the male protagonist, rather than that they're fully developed people. I also sometimes think his work comes across as male fantasy. I noticed it the most in Vicky Cristina Barcelona, but I've taken issue with it in other films as well.
How are his films unique in current cinema offerings, in your opinion?
I suppose that's several questions really. There are his older films which can be considered in the context of their contemporaries, as well as today's cinema; and then there are his newer films which can be looked at in the context of other current cinema. I think a blanket answer is that they (usually) don't talk down to their audience. I think Allen tends to be fairly self aware: he knows that he's assuming a certain degree of literacy on the part of his audience (see post #2 for some of the allusions in his work) and that it can come off as pretentious. Sometimes he makes fun of that very quality. But he's not afraid to present a movie for adults: no explosions, chances of a happy ending are 50/50, and it may just be a lot of people discussing ideas. I'd say in today's blockbuster oriented landscape that's rather unique.
I'll just note that I haven't seen Blue Jasmine yet. I really want to!
Yes, many of the characters are rather morose, but to me there's a truth to some of that angst, questioning one's calling in life. Maureen Stapleton adds such a joyous spark to the film, though, by revealing one of Woody's perpetual themes evident in several of his films (sometimes explicitly, like in Everyone Says I Love You: "Enjoy yourself; it's later than you think / enjoy yourself, while you're still in the pink..." 
lol
:
:
Purple Rose of Cairo, Annie Hall, Deconstructing Harry, Crimes and Misdemeanors and Midnight in Paris.
