I'm not equating it with narrow mindedness, I'm just relating the two. Everyone is closed minded to some degre, myself included. No one is completely open minded.
To answer your last question it varies. Three years ago I would not have been able to watch and appreciate Madoka Magica despite being an Anime fan already. I still can't appreciate DBZ or Naruto. I can't stand them. It can take years of gradually opening your mind to learn to appreciate a particular nieche depending on how much a person has closed their mind to it. I was initially repelled by Madoka, but I said to myself there had to be something worthwhile there for it to have so much critical acclaim. So I gave it another go and actually started liking it.
I am the same. We all have preferences regarding genres, however, these preferences themselves are not innate but they evolve with experience. Hence, there is something called being open to trying stuff that you usually don't experience and try to understand how some people appreciate it and them you will perhaps learn to appreciate it in the same way. That requires certain humility from the person as well to know that you might learn to enjoy things.
All my favorite things are to a certain extent the product of acquired taste. In fact, I think that everybody's favorites are, since you will only enjoy The Godfather as much as movie fans usually do if you have a substantial experience with American movies. If you show this movie to a polynesian tribesmen who never watched TV in his/her life, I don't think she/he will like it at all. You will also tend to only enjoy some otaku anime like K-On! if you have substantial experience with otaku culture. Some things are easier to enjoy without particular cultural experience because they are more similar to mainstream stuff, a movie like Wolf Children is much more similar to typical Hollywood drama movies than the Utena movie, hence people who don't have substantial experience with anime but have experience with Hollywood movies will tend to enjoy the first much more than the second.
I understood some of those things better in fact by interacting in this forum as well as other forums about movies, animation and other mediums. One can even say that I used these HoF to make some experiments regarding people's capability to digest otaku cultural products without being familiar with otaku culture itself. I learned a lot in the process.
I particularly didn't like K-On! the first time I watched it, I found it a bit weird and didn't quite notice what was so special about it, as it consists of watching a bunch of stupid highschool girls acting like retards. I watched it because it was named one of the best anime of the year in a magazine. But after watching all the 39 episodes of K-On! and K-On!! plus the movie plus about 30 other slice of life comedy cute girls doing cute things TV series and OVAs I recently rewatched the first episodes of K-On! and I understood why they were so popular among otaku, it's just a brilliantly made masterpiece of slice of life anime comedy. But it certainly is a very culture specific thing. Though comedy usually is.
Serious drama tends to be more accessible to foreigners than comedy, which explains why many people here loved movies like Princess Mononoke (being highest ranked foreign film in the 1990's countdown and top 10 in the animation countdown) and Sansho (even winning a HoF), even without being familar with Japanese culture in general. Comedy tends to be harder for foreigners to understand, I myself cannot understand what is so great about Coen's movies like Fargo and No Country for Old Men, being sarcastic tragicomedies that are very cultural specific to Americans.