A few points:
1) "(As gender is a social construct.)"
No. Masculine and feminine are not social constructs. They are the result of humans being quite sexually dimorphic. There are significant differences due to the influence of male and female hormones. What is a social construct is the rigidity of the gender roles. That is where it gets artificial, going over/above the natural differences.
Rigidity = "Boys don't cry", "Men can't enjoy cooking", "Men can't wear pink unless they're gay"
Natural = "Boys like action video games and girls like games where they dress avatars in a variety of neat outfits", "Boys fidget more, especially in school"
I've seen more than enough evidence of that natural example that I am certain it is more than just social programming, although it is true that boys are programmed to "behave like boys", too. Work in an elementary classroom, though, and watch what the two sexes choose to do with their free time when they have iPads. The vast majority of girls choose, without anyone prompting them, to do the avatar customization stuff and the boys choose the action games — even if they're playing solo with no peer involvement. And, boys do fidget more than girls. I've seen it time and time again. They want to be up and moving around. Girls are, as a rule, much more content to be less physically active. Girls are generally more socially polite and thoughtful in elementary school, too (with exceptions, of course). The differences are too common to be all due to socialization.
Of course, there is plenty of overlap between gender stereotypes. However, there are real differences between typical male and typical female behavior based on sex hormones.
2) "Androgynous". Androgyny should be looked at with a grain of salt. Except for chromosome abnormalities, like XXX females and XXY males, true androgyny is almost impossible to find. Now, an XXY male is the closest one can get to it. Also, often enough, the more androgynous-looking men are not more attractive. To me, the bodies of XXY males can have such womanly body shapes that I don't find that attractive at all. It's a matter of taste but masculine males are more common because that's what more women have found attractive enough to reproduce with.
Makeup, hormones, and plastic surgery can make people look more like the opposite sex but that's not the same thing as being naturally androgynous. I've seen some people who are quite androgynous without being XXY or XXX but it's rare and not necessarily more or less attractive, merely different. Being androgynous does not make you more beautiful by default. I know someone who could pass as a lesbian if he were to make some adjustments. However, he is neither as beautiful as women I would call beautiful nor as beautiful as men I would call beautiful.
3) The sexist idea that androgynous males are more beautiful than masculine males. This is based on the heterosexual male point of view that the female body is more beautiful. So, based on that bias, the male that looks the most female is the most beautiful. That is a bias, not a universal fact. Erasing the masculinity erases the maleness, so you're just saying that females are more beautiful in the end. Not everyone shares that particular bias. I was not particularly impressed with Paizo's Arshea due to this. The implication seems to me to be that male beauty has to be "males who look more like women than men" or "males who look really female, androgynous, are the most beautiful". That's either a heterosexual male point of view or the point of view of a transgendered person, or a heterosexual woman, or something. It is not the point of view of most gay men, hence the usually clear dichotomy between bara and yaoi. In Yaoi, males who look more androgynous are considered more attractive. In bara, males who look masculine are considered more attractive. Different biases. Bara males, though, are more true-to-life. Yaoi males tend to be rather alien-looking, stylized.
It is true that people (at least a statistical random sample in America) tend to see male faces with larger eyes, smaller noses, and other features more typically associated with women as being more attractive. Some of this (small noses, for instance) can be cultural bias. However, there is a big difference between being male-looking and attractive and simply looking like a mannish woman. It's complicated to explain but it's too simplistic to simply state that "male faces that look more feminine are more beautiful".
4) "Elves are historically depicted as androgynous." Elfin features are actually representative of a genetic disorder called Williams Syndrome. Whenever you read about the syndrome the faces are described as elfin. I bet folklore came up with the elf look concept, in part, based on this syndrome. The notion that elfin faces are more beautiful than regular human faces is one that I don't agree with. I was just looking at the drawing of the elf and half-elf in the 1e book yesterday and I didn't think either of them looked particularly gorgeous, especially the half-elf male with his bug eyes. I've seen plenty of Williams Syndrome faces. If you ignore their teeth (which are usually not well-formed) I don't think you'll find that their facial shape is more beautiful than a non-Williams non-elfin face. In fact, it's just unusual.
Male faces, in particular, tend to be wide, "flared", especially around the cheeks as I recall — and compressed in height around the eye area. The compression in particular seems to be the most common thing with male faces. The flare/compression runs contrary to the narrowed stretched elfin look. However, it is interesting that the 1e book's half-elf did have a wide face, although with a heart shape jaw.
5) "Elves switching sex is uncomfortable." I can understand why. A person's sex and gender are very strongly related to their core self-concept. I don't think it's transphobia, as someone suggested. I think it's mainly due to it being difficult for us to relate to a creature that doesn't strongly feel/see one sex/gender or the other as being strongly-linked to their core self-concept. It's like a creature that can switch from being old and young when it wants to. That is difficult to relate to. Elves are already quite difficult to relate to because of their very unhuman relationship to mortality and aging. Gaming/gamers have glossed over this difficulty a lot in the way elves have been depicted and behave. In reality, any creature that's as intelligent as an intelligent human that lives that long is going to act very differently than we do, at least the vast majority of us. Mortality, and the brevity of our lives (especially at our physical peak) plays an extremely important role in influencing how we behave and see the world.
Sex/gender switching makes elves even more difficult to relate to, more supernatural. The more supernatural something becomes the tougher it is for us to understand it and feel comfortable with it. The distance between us and it increases the more supernatural it becomes. This isn't a matter of phobia. It's a matter of basic intelligibility. One can more readily understand what is the most familiar. This is why men understand men better than women do and women understand women better than men do. It's why the concept of peers is important. Peers are more similar because of age and such, and thus typically easier to relate to.
Now I have to read the remaining 54 pages.