Intuition is still quite mysterious even to scientists but we consider it as our inner voice or gut feeling that is in contrary to thinking and reasoning. My guess is that our intuition on what's right and wrong is a psychological adaption powered by natural selection.
I may anticipate one of your arguments would be that ultimately human cooperate to increase their chance of survival which is a self-interested motive. Please note that the neo-cortex of the human brain which is responsible for rational decision making was only recently developed in the course of human evolution. Humans needed to cooperate in order to survive way before that happened. So my hypothesis at the moment is that the tendency towards reciprocity and cooperation is not only rationalized by our neo-cortex, the new brain but also rooted into our old brain which is responsible for our instincts.
Cooperation with others for survival is nothing new and exists in most animal species. It is not uniquely human. Example: A heard of wildebeest will attempt to fend off a lion attack. If left alone, they know their individual survival is unlikely. This also goes for the lions. A lone lion would probably be unsuccessful in an attack on a herd of wildebeest. This is why lone lions tend to either pick much smaller prey or resort to eating carrion.
Another factor in the animal kingdom as it is with humans is the rank within the species. The higher rank you are, the more resources you have. Those at the top (alphas) get the most to eat, the most opportunity to mate, and have more protection from the rest of the group. Those at the bottom eat last so may get no food at all. If resources grow slim, they either leave the group or die. Because they need to move up the ranks in order to survive, this places completion within the group.
A related factor is the presence of other groups of the same or other species. Individual groups depend on their resources. If resources become too slim for the group to survive, they must expand their territory. This often lead to competition with other groups over the territory. The defending group may also need this territory to survive.
Though cooperation may be necessary for individual survival, competition within the group and against other groups is key to self preservation.
Cooperation had been a long time goal around the world. In some regions, force was applied to unify the region such as Japan but others came out of negotiation such as the U.N. The issue with multiple countries cooperating has always been "Who is in control?" Obviously, no country would give total trust to the ruler of another. This would lead to the fear that their country would loose it's power as a result and be put in control of the foreign leader. One of the things that history has told us is that alliances are temporary. People who are your allies one time can be your enemy another, and those that are your enemy may some day be your allies. Knowing this, is there any wonder why mistrust between nations is so common?
Right and wrong depend on what favors the well-being of ourselves and of others. Factors relating to that can explain why our moral intuition tends one way or another but it's an inescapable fact that sometimes this intuition turns out to be wrong. In fact if it was never wrong we wouldn't reason or think about "morality" ever as there would be no need for it. Of course people sometimes think and draw the wrong conclusions too, so reason isn't foolproof either. But if there was anything that was foolproof we would live in a utopian world.
As for cooperation v. Competition. Is it morally right to cooperate with Hitler? Unfortunately many people at the time thought so. I think that the tendency for most people to assume that cooperation is right and competition is wrong for a given case is part of what enables tyrants. This isn't to say that cooperation can't be right and it often is. Cooperation can be beautiful as long as those cooperating don't surrender their ability to think for themselves.
Some people, less people, but some people will more instinctively compete, they will more intuitively think "I'm not sure if this is a good idea" they will come up with counterfactual ideas which sometimes work out better and sometimes not. It is good that we have these people. They put the brakes on slides into tyranny. Perhaps if one such person was able to understand the German public at the time and sway them away from Hitler things would've turned out better.
To have a good grasp of morality you have to recognize both the value of cooperation and competition and use your best judgment to decide how to cooperate (and with whom) and how to compete (and against which ideas).