I'm no expert, but as far as I know 'see' is not usually used in present progressive: http://www.englisch-hilfen.de/en/grammar/present_progressive_verbs.htm
Maybe that's why it's not accepted.
Correct. When we use "seeing" it is almost always with the meaning of "dating": He has been seeing her for a couple of months. הוא יוצא איתה כבר חודשיים.
Not exclusively. It's also fairly common in terms of movies, shows, or concerts. Like "They're seeing Madonna tonight." Or "We're seeing X-Men." But in that case you're generally also answering the question "Who are you going to see?" Or "What movie are you seeing?" But yeah, between your example and mine, I think we've basically covered the only grammatically appropriate uses of "are seeing". So dating and entertainment, I guess.
Yeah, the whole issue of "stative verbs not being used in the progressive" is a bit silly, because there are so many exceptional cases where you can use them in the progressive. I'm liking your new outfit, i'm lovin' it, are also examples of that. The main thing about them is that they can be used in the simple present without it meaning something general or repeated, unlike "normal" verbs: "I play tennis" cannot be used if you are doing it at that moment in time, it means that in general you play tennis, whereas "I see the ball" can mean that you see it at that moment in time.
Answering in general. What are you doing tonight? Where is he taking you? We're seeing this band tonight... We're going to the movies.
(Any plans to go to concerts at (name of a music venue) this summer? Tonight we're going to XX to see....