(This was the Fuck Yeah Glee Podcast about the Berry, Puckerman and Fabray families.)
I'll work from the limited understanding a semester of African-American history and independent reading has given me, to summarize a little history. There were various racist laws classifying African-Americans by the mix of African and European ancestry they had. Laws were different in different parts of the country, and some conflicted. In taking censuses, the census taker might be the one who made the call. The words for one system of these old classifications partly came from Spanish words. "Negro" was "black." I believe "mulatto" was from a word for mule, which is among the reasons the term is outdated now, or at least really depends on who's using it and why. Saying it was a historical term is one thing, but it's not a word to use for people in the modern day. "Quadroon" and "octoroon" are probably from Latin words. "Octoroon" is still used today, but again it really depends who's using it. Someone who is that can use it to describe themselves. But that system of classification went all, half, fourth and eighth for African descent. To my still limited understanding, parts of the country that were Spanish and French colonies used this, although it later became more widely used.
There was another law, the "one drop" rule (http://en.wikipedia.org/wiki/One-drop_rule), that said that anyone with any African ancestry at all was Black. These laws are no longer on the books, as far as I know, but still had influence decades later, at least through the 1950s. To at least some extent, the African-American community accepted anyone with African ancestry. Times have changed, and different people have different opinions in different places. But there are very light-skinned people who identify as Black, much lighter than Jake. This is why I say that in real life, if Jake identified as Black, he would probably be accepted as such. (I am talking about the character.)
Rachel not knowing if she were mixed race was played as a joke. With the model in the picture, the joke was that it would be obvious if she were biracial. With the actor who played Leroy Berry being lighter, and how much Rachel takes after Shelby, Rachel could have been multiracial. Rachel never got checked on her white-skin privilege, though. As I posted, if I remember correctly, the discussion at the time Leroy showed up in the flesh was that the actor who played him was multiracial: Black, Native American (and white?). Leroy didn't "change race," though. Leroy was established as Black, and that didn't change. Rachel not knowing whether Hiram or Leroy was her biological father didn't change.
Glee kind of does go into the social construct aspects of race. Not as much as it could, but it's there with Rachel, and with Jake. Rachel doesn't question her privileges. Jake gets discrimination from all sides.
Added: In the episode "Sweet Dreams," when Marley was talking about her original songs, Kitty said people didn't want to hear "a song about loving an octoroon." I figured that since it was Kitty, viewers would know she was being offensive. Jake did get a reaction expression, and he didn't look too happy.