Yes, homosexuality does go against many cultures. I'm not trying to argue that homosexuality should be universally held as morally acceptable, even though I believe that would be good, but rather that homosexuality wouldn't slope towards beastiality, and thus your argument by analogy is wrong. Another Humian argument would be that an analogy is only as strong as it is similar to its comparison. The more similar, the stronger the argument. Your analogy of the gay marriage leading to interspecial marriage (I guess that's a word??) doesn't particularly resemble mine on eugenics, and so isn't a successful refutation of the slippery slope. I would say, however, that my analogy of Nazi Germany's eugenics compared to abortion eugenics is fairly similar, as both systematically seek to eliminate certain characteristics.
If you're still not convinced by mah boi Hume, how about this: you could just as easily state that heterosexual marriage slopes to interspecial marriage, which there is no evidence for. As gay marriage is a very good analogy to heterosexual marriage, you'd expect the same to be the case for gay marriage. However, I doubt that it's well documented this is the case (there is a substantial collection of married people, I'm sure the correlation would have been found ages ago if it existed).
I suppose it is consent, but I'm also sure that animal would be doing that solely based on instinct to procreate. To take advantage of that for your own pleasure, I feel, is wrong. Furthermore, you can't know when the animal is in pain easily, as not all animals are very vocal. This is one reason why beastiality has recently been illegalised in places like Denmark. However, your argument about animals not being able to consent to being owned is pretty strong, though I still feel it's becoming irrelevant. I could counter by saying that, from a Utilitarian perspective, keeping animals keeps them happy (as they are warm and well fed hopefully!), and owners happy (the joy of having a pet, along with research that stroking animals helps reduce heart disease etc.). Overall, it could be said to be in the best interests of both. Beastiality, however, may hurt the animal, and so isn't justified.
Yes, the slippery slope argument works against me, which is partially why abortion is such a difficult topic. The line is very blurred. However, abortion on the grounds of quality of life, I think, is a fairly uncontroversial view. It's about avoiding unnecessary suffering, not seeking to avoid having people who are different just because they don't have the same mental faculties as most (which I assume is your reason for wanting to abort DS foetuses, because it certainly can't be on grounds of poor quality of life). I think this is where intention is key. Intention to conserve quality of life is not what concentration camps in WWII were based on, but intention to select those who don't appeal to some ideal standard was.
However, it may be the case that the UK is undergoing a slippery slope. "Between 2011 and 2013, there has been a 17.8% increase in the submission of HSA4 Abortion Notifications for Down’s syndrome. The Department of Health will continue to work closely with RCOG and other organisations in implementing the recommendations where possible." From https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/433437/2014_Commentary__5_.pdf
This is a government report, so I'm pretty sure it takes an objective stance.
However, you could refute this by arguing that, as gender equality seems to be on the rise in the UK and more women are professionally active, this is because women are generally having babies later. Later pregnancies lead to an increase in rates od DS, as the eggs are exposed to more net radiation and so are more likely to mutate.
Then again, it may not be this factor. I'm sure there are a huge number of socioeconomic factors which play into this alone, so it's really hard to isolate one and state it's effects without cherry-picking. Therefore, I don't know if the UK is sloping, and I doubt you do either. Again, this comes to my hedge-your-bets argument (which isn't a particularly good one but is, I would say, the safest approach to the issue just in case it goes balls up and does form a slippery slope). I have seen this 90% abortion rate you said with a quick online search, but only from what seem to be fairly opinionated sources. I'm a bit sceptical about them, so I'm having a look on the NHS website.
"However, improvements in the screening process has led to a decrease in the number of Down’s syndrome births due to a high rate of terminations. The study also suggests that the rise in Down’s diagnoses is associated with a rise in the number of women putting off having children until later in life.
Antenatal screening and subsequent terminations of pregnancy resulted in an overall 1% fall in the number of babies born with Down’s syndrome (752 in 1989-90 to 743 in 2007-8). This equates to a decrease from 1.10 babies per 1,000 births to a current rate of 1.08 babies per 1,000 births. The research estimates that if screening had not been in place the actual numbers of live births with Down’s syndrome would have increased by 48%, from 959 to 1,422. This estimated rise appears to be attributable to the fact that couples are starting families at a later stage in life."
This is from the NHS choices website: http://www.nhs.uk/news/2009/10October/Pages/Down-syndrome-termination-rate-screening.aspx
So nothing on the actual stat of those born with DS : those terminated, which I'm a little surprised about. Who knows if the UK is sloping? I hope not!
Your burden argument also might apply to any disability that requires social funding by it's own reason. That's another reason it might become a slope.