I think one problem with looking at it so dispassionately is that it makes it harder to see one's self as being in that position while such consequential issues are being decided over your head. I think that the fact that we are talking about people and not chairs requires a less detached view.
While I do view it as an ultimately moral question, I think this eugenics thing is, in it's simplest terms, allowing dramatically important decisions about your life to be made over your head by people who do not have your condition and are making general suppositions to justify taking one position or another. That sounds like a bad idea waiting to happen. Yes, the same argument can probably be made for surgery but genetics is far more complicated than surgery and surgery is plenty complicated.
But to answer the questions:
Each time we use the "morality" argument first of all it should be good to define what morality is, why we think it's good and what consequences it has.
There is no one definition, so you can't necessarily "define" it like it is simply a word. Morality will vary from person to person, and sometimes, from person to person it will be a contradictory term. But, regardless of what label is put on it, we can more or less agree on the fact that there is concern as to the outcome of Eugenics likely on the basis of compassion towards fellow humans.
Speaking about eugenetics, think to diabetes: why people think that curing diabetes with insulin injections is morally better than curing it (supposing for this example that's scientifically possible) with eugenetics? In the long term insulin injections have many side effects and I know about people living an almost normal, but painful (loss of limbs and sort of dementia* induced by liver disease caused by the cure) life.
Producing insulin and injecting poses less of a risk than playing around with genes that are still in the process of being researched. Think of tampering with genes like it is archery. You can stick a bow and arrow into anyone's hands and make an attempt at hitting the target dead in the center; however, it is only the experienced of someone that knows what they're doing that can increase the likelihood that the target will be hit in the center with as few injuries or casualties as possible. This is not to say that it would be impossible to one day safely tamper with the genes, but even using thoughtful and careful medical gene therapies can be risky now.
If our judgement shouldn't be better than the nature's one, why we practice medicine at all?
We practice medicine because of compassion and need. We cannot stand to suffer any more than necessary and we, hopefully, cannot stand to watch others suffer. However, there is a difference between coercing someone into being sterilized and giving someone a medicine that may or may not harm them when all other options have been exhausted and they have full disclosure. While the same arguments against Eugenics can be used against medicine, it is not always an equivalent scenario.
If we live following nature's rules you can be sure a lot of diseased people will not reach an age suitable for reproduction, Down syndrome included.
Without special intervention all people regardless of health can potentially die from a variety of issues even if they are very healthy and strong. Think of a tsunami scraping everyone out to sea and drowning them. But, as soon as you start tampering with your life as it would be without technological and societal advances you are not "following nature's rules". Once you have medicine, start changing the very nature of the food you grow, try to change or resist the weather, so on and so forth you are not living by "nature's rules" and it seems out of place to try to bring nature back into the discussion. Either you want to live in simple dwellings subject to whatever nature throws at you or you want to try to "better" your situation. Then the discussion comes full circle back to morality and definitions of "better".
Should we put in discussion if our morality is totally fault proof?
I'm not sure what you mean here.
Is our morality heavily influenced by religion?
If so, how many lives have been saved by prayers?
Even when morality is influenced by religion, there is more to the discussion than "can the illness be prayed away?". If you have a higher power that says you must eliminate the weak and preserve the strong, that can serve as the basis for a morality. The same is true if a higher power says the strong must protect the weak. For any religiously based morality, you must take it, as all the other definitions, on it's own specific merit as religions cannot accurately be lumped together any more than people can.
Yet, for what it is worth there are people who claim to be saved by prayers and always have been since before we developed sophisticated medicines, and despite how harsh the world can be, some people survived long enough for us to get here and to develop medicines. The issues remain, however, "what standard of morality do you use" and "how does it apply to eugenics."
What today we think as being morally controversial perhaps in the future will be simple routine.
Think what Demikhov did: it's horrible, but probably without those experiments even transplants could be thought to be impossible today.
Agreed, it's horrible, but the wiki at least only mentioned him working on animals, not humans. While I don't like the idea of testing on animals, that is a completely different moral argument. Yet, if he were working on humans and not animals, I would rather not have access to the medicine we have now than subject someone to stuff like that. For instance, modern medicine apparently owes much to the Nazi's but given a choice, I would rather do without than have those poor people experimented upon as they were.
