Solaris1

I'll attempt to defend that there are NO genuine moral disagreements. 

I'll attempt to defend that we all agree with the following principle:

Any reason would appeal to other's human rights. 

So.... Is there any counterexamples?

One possible one are Nazis, they believed we could violate the human rights of Jews. However, this disargeement is about reality, not ethics. The Nazis and I disagree not on the principal above, but one whether Jews are people. So there's no moral disagreement.

The interaction problem is a supposed problem for Dualism, the view of the mind that humans have two parts, an immaterial mind and the body

I will state it as it follows:

1-The mind and body are two separate substances, and have no shared properties

2-two substances need one shared property to interact

3-the mind and body cannot interact

However, the mind and body do clearly act. Whether you are a materialistic, Idealist or whatever, you most likely belief that your thoughts cause your actions. You need to drop either one or two. 

Two can be supported by the fact it has wide confirmation: a hammer and a nail have the property of being physical, and ideas that interact with each other have the property of being ideas. 

Do you have any evidence/arguments that would prove that humans have souls/immaterial minds?

Evidence against the immaterial part of the mind is Phineas Gage [1].

Popular reports of pre-accident Gage often depict him as a hardworking, pleasant man prior to the accident. Post-accident, these reports describe him as a changed man, suggesting that the injury had transformed him into a surly, aggressive alcoholic who was unable to hold down a job.

Damage to the Brain = damage to the mind, therefore mind is not separate from the brain. 

[1]

The problem of evil is an argument against the existence of a God. Therefore, it sets out to prove a God does not exist.

Skeptical theism, as a response to the problem of evil, states that we cannot see God's intentions and therefore cannot claim the evil in the world disproves God. At the very most, we may be agnostic towards the idea that God has good justifications for allowing evil.