In the past 2 years or so I've became a big Japanese anime fan, and I've probably watched 50 movies by now (the only series I watched was Neon Genesis Evangelion). In every movie I watched (and NGE for that matter) there was always some scene in a train - which is way more common than in other genres. I've read about how advanced the trains are in Japan, and I can accept that the average Japanese guy uses the train more than non-Japanese people, yet most of the train scenes are more than just "traveling" scenes and they are usually very emotional.
As senshin mentioned, train rides are a huge part of the average Japanese person's daily life. Scanning the average daily ridership column on Wikipedia's list of urban rail systems in Japan, you can see that most of the lines are somewhere in the 500,000 range. Tokyo's busiest system, the Tokyo Metro system, carries around 6,000,000 passengers per day across all of its lines. Comparing that with this list of rapid transit systems in the US, you can see that the busiest is New York's subway system, which carries around 9 million passengers per day, but the numbers rapidly fall off and the San Francisco Bay Area's BART system, in fifth place, is only in the 400,000 range. As for US light rail, it starts in the 200,000 range and falls off from there.
As to your second observation, I don't know if you meant something specific by "very emotional", so if I've missed the mark, please do clarify. The most obvious reason why train scenes are more than just traveling scenes is that traveling scenes are bad writing. I ride the train every day, surrounded by a thousand other half-asleep people staring at their phones. It's quite boring. Reproducing that in anime is not a good call, especially when you're making a film and have limited runtime in which to tell your story. So if there's a scene on a train, it's going to advance the story somehow, and emotional scenes are much easier to do on trains than fight scenes or infodumps on classified government projects. In my experience, anime films (that aren't based on an existing series) tend to be more on the dramatic and emotional side anyway, so it's likely that many of their crucial scenes will be emotional ones.
Sometimes the train also serves as symbolism for something; this page suggests that the train in Eva symbolizes Shinji's way of dealing with the world. (Trains are definitely a place where people have their AT fields up at full strength.) The authors set a scene on a train because the fact that it's in a train makes some subtle comment about what's happening in the scene.