According to this, does someone who donates sperm violate the prohibition of wasting semen? Or since the sperm is intended for fertilization, even if in the end it may not actually succeed, is it not considered wasted semen? And another question: assuming there is some prohibition involved in sperm donation, does that mean fulfilling the commandment of being fruitful and multiplying in this way would be considered a commandment that comes through a transgression, and therefore would not count as a commandment fulfilled?
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I think it really is not considered in vain. Even if halakhically he would not be considered the father, I am not sure this would count as wasted semen. It is producing a child who will be someone else's offspring.
If the commandment belongs to the donor, then this is a commandment that comes through a transgression. If the commandment belongs to someone else, who is the father, there is room to argue that it is not. But as I said, in my opinion it is clear that the donor is the father.

In your opinion, is there any prohibition at all in sperm donation? Is it something proper?

Clearly it is problematic. The semen is not wasted, but there is adultery by hand here and forbidden thoughts. Therefore, even if people donate, it should only be because of the distress of a couple who cannot have children. And even in such a case, in my opinion it is preferable that a married man do this with his wife.
One has to check how this is actually done in practice and clarify it with people who are familiar with the facts and the constraints.

I'm asking because on the one hand you said that someone who donated has fulfilled the commandment of being fruitful and multiplying, and on the other hand you said that if there is a transgression involved, then he has not fulfilled it because this is a commandment that comes through a transgression. So how does all this fit together?

I didn't understand the question. To your question I answered that if indeed there is a transgression here, then it is a commandment that comes through a transgression. But there is no transgression here if it is done in the proper way and for a genuine need.

Let me ask differently: if someone donates sperm in the usual way, meaning not a married man doing it with his wife, can he thereby fulfill the commandment of being fruitful and multiplying? Or since it involves adultery by hand and forbidden thoughts, is it a commandment that comes through a transgression?

If there is no one else who can do it, then perhaps it is permitted for him for the sake of helping a needy couple, and then there is no transgression here.

But practically speaking, there is always someone else who can do it. And furthermore, donation is usually done anonymously, when there is no connection at all between the recipient and the donor.

In that situation it is certainly possible that this would be a commandment that comes through a transgression. It depends on the parameters of that principle.