So, in hungarian, we call the the branch of history that deals with rulers, how long they ruled, what treaties that made, whom they were succeeded by, what wars were closed by what peace agreement, etc, "Politika történelem" which would translate in to "Political history". Would that be the right word for it?
Also, when History is studied through the perspective that is also the purview of another science, like economics, sociology, demographics, we call that a "segédtudomány", which i think translates to "Auxiliary science". For example, in this context and in this perspective, if you study the economy of let's say medieval france, Economics is used in that case as an auxiliary science of history.
Are these the right english terms for the things I described?
On the first question, yes, political history is the name given to the happenings of states, and all of the examples you provided would fall under that term. Political history, it's worth noting, was long considered the established, classic history.
In the United States, there is not a commonly used phrase that encompasses all specific branches outside of traditional political history. You might hear "non-classical" or some other ad hoc term to refer to historiographic approaches outside of the traditional mainstream, but with respect to a direct translation of segédtudomány, it doesn't sound like there is a similar term. Rather, you would refer to specific approaches/book/etc. as economic history, social history, etc.
If you're curious about learning more, you might check out "The Modern Historiographic Reader", which is a series of essays on, you guessed it, historiography. I haven't read each essay, and some are far more readable than others, but it will give you enough to grasp the basics and nail down terms.
I'm familiar with "auxiliary sciences of history" in the particular context of medieval studies. "Auxiliary sciences of history" refers to the particular skills and disciplines that are facilitate historical studies. These include paleography (the study of old handwriting), diplomatics (the study of official documents), numismatics (the study of old forms), codicology (the study of old books), chronology (the dating of events), sigillography (the study of seals), prosopography (the study of groups of people; "collective biography") and a number of others.
So to pick a very basic example, if I want to access the information in an 11th century manuscript to assist my study of medieval political history, paleography is going to be involved in order to read it. Maybe the historian is reading it. Maybe the historian is relying on a printed edition prepared by someone who did.
The examples I gave are perhaps clearer than economics because they are not full-fledged academic departments and are generally seen as very specialized and dedicated to assisting and facilitating. But other academic disciplines may also be considered "auxiliary sciences" if they are put in the service of historical studies.