I have an idea which would solve every problem on earth.

I know that, and I also know that because of that, the opinions about it will be mostly unfavourable. But just because in the past people have done unethical things in name of eugenics, doesn't mean there is no argument to make about it's potential.
Also the eugenic enthusiast in the past weren't focused on creating more empathy, but were focused on people with low intelligence and abnormalities in people. But clearing the world of unintelligent people won't make the world necessarily a better place.
So I think there is a big difference there.

You can't objectively measure empathy yet. That's why I said we'll probably going to need a couple decades of research. There has to be a 99.9+% accuracy on the lie-detector test for example to be able to use the result.

You say you can't measure empathy, but I think it really should be possible to do so, because since I can clearly distinguish when I'm feeling empathy or not, there simply has to be some measurable indications on the outside of this feeling. We just haven't really tried yet to measure empathy.

I'm of course talking about real empathy not fake empathy.
You ask me what makes people with high empathy superior, but I think I already answered that question in my first post. If everyone would have high empathy, then who is gonna hurt others, who is gonna kill others, who will wilfully destroy the earth? The point is that everyone will feel repulsed by these sort of actions, while there will be an abundance of good actions, ensuring the safety and happiness of mankind.

Telling people to not have kids is rough, I agree. But I didn't say they can't raise kids, just preferably not their own. The reason I say preferably is because I don't think you should force unempathetical people to not have kids, only discourage it, and reward it if they don't. While this might be harsh, the effect it will have on the future I think outweighs the down-sides.

You’ll never be able to objectively measure empathy until mind reading is possible, and probably not even then. What a person is feeling is completely unobservable or recordable in any way, and often we struggle to even put into words and describe our own feelings. We just don’t know even close to enough about the brain for it to feasible any time soon, and for all we know it might never be possible. Same thing with a “99.9 percent accuracy lie detector test” no such test exists. The closest is polygraph tests, which barely did better than 50 50 chance. There’s no current accurate way to measure such a thing.

There are no objective outside signs of what a person is feeling. You alone control what you want people to see, and body language isn’t always indicative of what a person is feeling. When you say that people with empathy are repulsed by bad actions, you mean they are repulsed by what they consider to be bad. Not everyone has the same moral compass however, so even in this supposedly ideal scenario where the world is mostly comprised of people with high empathy, these empathetic people might still do terrible things if they have a different moral compass to you.

Finally, encouraging people with low empathy to raise kids that aren’t their own would be redundant, firstly because the lack of empathy of their parents could theoretically rub off on them, as we don’t know for certain whether empathy is 100 percent determined by DNA or not. And secondly, in a world where all mothers are empathetic, wouldn’t the rate of kids left to be raised by a different mother plummet? Meaning basically the foster care system would become redundant in this most ideal scenario, which also means there would be almost no kids for these non empathetic people to raise.

As soon as DNA was mentioned I immediately thought of GATTACA

Basically it's a movie about a society where almost everyone is gene edited to be perfect. Those who are "faith births" (Not edited) are paid the bare minimum and treated like slaves, often doing societies dirty work, such as being a janitor

As you can guess, DNA tampering sounds bad

There's no proof that DNA is related to empathy at all. There's actually a pretty good book I read a long while ago, I don't remember the title or author but basically it asked what would happen if you took perfect copies of the worlds worst criminals and grew them up in a perfectly controlled environment with traditional "good" values promoted in their education, would they still grow up to be like their grown up counterparts or would they become model citizens? Pretty interesting question, and personally, I lean toward thinking that it's primarily based on how you are raised and what you learn naturally from your parents rather than DNA. I don't know for sure, but that's my perspective anyway.

It's probably a mix of both, but your future behaviors are probably influenced a lot by your childhood, and especially at very early ages when you're still developing. But I'm sure DNA does have a role to play as well.

That's the other issue