So, were there any documented cases of consensual interracial relationships? Sure. Especially in the 1600s there is strong evidence of consensual relationships between enslaved blacks and low class whites, but there were also many laws passed in the late 17th and early to mid 18th century intended to curb these relationships, especially between black slaves and white servant women. Strong legal disincentives to prevent these women from becoming a "disgrace of our Nation" passed throughout the colonies. In Virginia 1691, for instance, the law mandated that a white person marrying a black man (whether free or slave) would "be banished and removed from this dominion forever" and further that a white woman who bore a 'bastard' 'mulatto':

she [shall] pay the sume of fifteen pounds sterling . . . [or be] disposed of for five yeares ... such bastard [shall] be bound out as a servant ... untill he or she shall attaine the age of thirty yeares, and in case such English woman that shall have such bastard child be a servant, she shall be sold by the said church wardens, (after her time is expired that she ought by law to serve her master) for five yeares ....

Now, this is likely to what your friend is refering (although obligatory caution when discussing Irish servitude in the early colonial period), but there are clearly a few issues here. The strong genetic evidence of European ancestry in the African-American population does come from the male line, not the female line, in much greater proportion, and does also indicate that it heavily dates to the late-18th through mid-19th centuries (see this article by Dr. Henry Lewis Gates, Jr.) The simple fact is, and I am sorry to have to say this, that rape was incredibly widespread within the antebellum South. Yes, there will always be exceptions, and I can dig up more, and more recent ones from the 1800s, but that won't change that underlying fact, for which I would direct you to this longer response in the recent AMA. So in short, yes, consensual relationships contributed, but can by no means explain the extent of the admixture that these test results demonstrate.

Regarding Jews specifically, there are two main was Ashkenazi Jewish genes could've gotten into the African American gene pool.

The first is the obvious one, Jewish slave owners. While there were not a particularly large number of Jews in the southern US in the pre-Civil War period, there was a non-negligible minority present. Because Jews tended (both in this specific instance and in general) to be a fairly urban population, Jews did not tend to own either small farms with slaves or large plantations, so numerically the number of slaves Jews owned was not particularly large. But, Jews in cities might own a few slaves as domestic servants or to work in their businesses. And indeed there is documentary evidence that this was the case. Presumably in this context Jews might have children with slaves, either consentually or (sadly more likely) not.

The second is a little more subtle. Jews in Europe converting to Christianity was something that happened at various times and places, under varying degrees of compulsion. There were usually social and economic pressures to convert to Christianity, sometimes physical force (i.e. convert or die). A small number of cases of this, compounded over centuries, eventually ends up as being a pretty substantial level of population mixing. Anecdotally, it's not uncommon for people to have these genetic tests who are simply European to have some level of Ashkenazi Jewish ancestry, even if they have no recognizably Jewish ancestors in the recent past.

Additionally, the point /u/hillsonghoods also applies to this specifically. Untangling Jewish genetics from Middle Eastern and European genetics is not such an easy task. It's possible some genetic pattern identified as "Ashkenazi Jewish" is really just European, and through random chance (or the founder effect) is over-represented among Ashkenazi Jews. Or it's possible that the pattern is actually Middle Eastern.

DNA evidence has a lot of potential, but it's not as straightforward as your precisely calculated results suggest. It's especially tricky with regard to questions about ancestry. Basically, the assumption goes, there are large geographic areas where the genetic population has been stable and consistent deep into prehistory, so if we take a sample of someone's DNA, we can compare it to a database of other present-day samples, and deduce what geographic areas their ancestors came from.

Except this line of reasoning has a lot of problems. If your dataset is incomplete—say, for example, that some parts of the world like Europe are better represented than other parts of the world like Africa—then a DNA sequence that matches an unsampled part of Africa might register as a false-positive for Europe. So your results more precisely indicate how closely your DNA compares with an incomplete sample of present-day populations, rather than a snapshot of your ancestry.

But perhaps most importantly from a historian's point of view, the DNA testing of present-day populations doesn't indicate when genetic drift occurred. Geneticists have developed elaborate models to make educated guesses, but these are largely untested. For example, we know that Scandinavians were sailing off the coast of West Africa in the 800s and 900s (thanks in part to mouse DNA!). If these populations were unable to return home, they may have integrated with African populations, perhaps even contributing to your 2% overlap with Scandinavian populations.

Conversely, European DNA has also been diverging, particularly with recent migration from North Africa and the Middle East into Europe, as well as other places. Great Britain has, for example, been more deeply affected in recent centuries as a hub of empire, to the extent that curry has become something of a national food on the menu of any self-respecting pub. There have been some small studies trying to sort out English DNA from this post-colonial DNA (such as building a dataset using only people with traditionally English last names), but there's no guarantee that the datasets used representing all countries differentiate between 'native' DNA and DNA that represents more recent immigration. So your results might more accurately suggest that 27% of Europeans have at least some modest level of African ancestry, rather than the other way around.

And since you're concerned about the impacts of European rape, it's worth noting that the period of European colonialism in Africa took off after the slave trade had largely ended. Europeans mixed with local populations, in some cases consensually, and in some cases not so much. Since colonialism was often dominated by males, it is likely that this changed the genetic pool for male DNA especially in the colonial centers, which I suspect are over-represented in DNA datasets. So you may be getting a false negative for Africa because DNA pool in certain parts of Africa has drifted since the early 1800s, after your ancestors likely left.

Since DNA populations are not stable, to get accurate results about your ancestry, you would actually have to test your DNA against archaeological DNA. I suspect that if there were a full dataset of DNA for individuals buried around 1400 in both Europe and Africa, your results would look very different. And if you had a dataset from earlier, say 600 before the Islamic conquests or 300 before the fall of Rome, and your DNA might point in new and different directions.

Sorry if that's all a bit sloppy. I'm no expert in genetic research or on the history of the African diaspora. But I do know that historians and archaeologists have spent a lot of time wrestling with genetic evidence presumably associated with the post-Roman migration period and the Viking Age in particular, since the proxy populations from today are relatively well mapped and the historical moments in question have a lot of political impact and popular appeal. But we're just now beginning to investigate DNA in ways that should give us solid information about migration and population change at these particular points in time. Up until now, both our datasets and our interpretive methodologies have been deeply flawed—and I say this regarding major population movements that are fairly well documented and which have produced modern populations that are fairly well represented in DNA databases! For an overview of one project and the problems it's trying to overcome, I'd strongly recommend checking out: Mapping European Population Movement through Genomic Research (pdf).

With regard to your genetics teacher, I find it personally troubling that s/he is using flawed datasets and untested models of genetic drift to interpret the family histories of his/her students. Perhaps s/he is doing this to get students interested in genetics, but it's being oversold and potentially damaging. No class exercise should be used to encourage an identity crisis. Yes, we know that the slavery experienced during the modern African diaspora was bad. But your DNA test cannot be used as proof of any particular historical events, since the DNA inherited from your ancestors is being compared against small samples from the moving genetic pools of today.