Why has monarchy appeared just about everywhere?

I disagree with the notion of it not spreading.

Conquered territories need to be governed to the conquerers liking (If not, what's the use of it?) - in the time frame this question makes sense (I'm reluctant to put a year here) there were basically only dictatorships or primitive democracies (Ignoring e.g. tribal people or differences between the greek Agora principle and the roman consulate)
The term for this is client state or (in the persian case) Satrap.

It makes sense [citation needed] for the conquerer to force the usage of the government system he comes from.

So your actual question is why did monarchies appeared in vastly different cultures such as the cultures with a Sumerian, Chinese, Egyptian and Inca background.

And that is simply because actually before they were a culture but just a tribe there already was a leader. The appointment of that leader can have various reasons:

1. strength/intimidation
2. intelligence/leadership skills
3. good storytelling abilties/access to hallucinogens/contact with the spirits and gods.

Whether the offsprings have the same abilities is debatable (regression to mean and all that) but they already have the family bonus, an insight into the leadership and most importantly:

For instance, one man is king only because other men stand in the relation of subjects to him. They, on the contrary, imagine that they are subjects because he is king.

(http://www.marxists.org/archive/marx/works/1867-c1/ch01.htm#22b)

According to Elman Service (who coined the Band-Tribe-Chiefdom-State sociopolitical typology), societies "naturally" tend to centralization of power when they begin to do agriculture. One argument for that is that central power leads to more efficient promotion and coordination of large infrastructure works, in particular irrigation. The leader also uses his power to reward his followers, which strengthens his position. In that sense, it is expected that all civilizations (by which I mean "societies with cities") tend to become monarchies. Even historically, well-known non-monarchies (e.g. the Roman Republic) appeared after an initial monarchistic state, as a reaction which needed the invention of mechanisms and secondary powers designed to prevent the current leader from becoming a monarch (e.g. the two conjointly elected Consuls and the Senate).

As for the monarchy becoming hereditary, it can be viewed as a natural consequence of the combination of monarchy and inheritance. Once the principle of handing over your worldly possessions to your children at your death has been established, it becomes logical that the monarch would like his own children to inherit his power. Early reigns in most dynasties tend to show a power struggle between the biological heirs, and the representatives of the other clans who helped establish the current power. When the heir is strong, the dynasty is established and the principle of hereditary monarchy becomes established; otherwise, the monarchy remains elective. An example is the Capetian dynasty, which was initially elective: kings were taking care to associate their heir to the throne (with a crowning ceremony) so that the heir really inherits the throne. It took more than a century for the French monarchy to become really hereditary, i.e. such that one descendent of the previous king would be considered to be the new incumbent even if there was no prior association.

To sum up, societies tend to become monarchies when they become complex enough to support cities; and monarchies tend to become hereditary because monarchs work hard to make it happen that way (and they do so following a tendency to favour one's children, tendency which appears to be quite universal).

(As will all theories about sociopolitical history, all of this is much debatable, and debated.)