These questions ought to be reworked as individual historiography questions on main. However:
What is the difference between history and politics?
What is the difference between history and sociology (or anthropology, or any other related disciplines)
"Large-n" discursive analysis of documents ("texts") in light of a tradition of reading-and-writing in a genre called scholarly history. Theory is optional, but by preference is concealed beneath the narrative. The unique element of historiography, as compared to other social science or humanities disciplines, it the attempt to read with veracity evidence of the past from very large collections of documents.
What are the distinctive methods of history?
A unique kind of reading and then writing.
What is the difference between a professional historian and an amatuer?
A PhD, or positive reviews in a highly esteemed journal of a key monograph, or the publication of six or more papers in a highly esteemed journal. Amateurs "tend to get on with it" at the cost of writing about irrelevances in poorly thought manners without an adequate source basis. It generally isn't the selected subject matter. I've been looking at local women's organisations and civic organisations as a potential area of interest. But there's a gulf between my desire to historicise cross-class local provincial organisations compared to a list of names and local figures isolated from the context of mid-Twentieth century associationalism.
What is counterfactual history?
Inverting a commonly accepted causal structure to test its causative properties by speculation.
Is it merely applying legitimate historical methods to flawed and erroneous propositions? (e.g. The Great Game)
It is more of a null-hypothesis test. See, for example, Brooks' Sealion ( http://www.philm.demon.co.uk/Miscellaneous/Sealion.htm )
How do we study history?
Ideally, do a BA Honours (Australian style) or BA+MA (Bologna), with a double major in History and a Language. Then do a PhD. Then start reading and writing based on further projects (the PhD ought to teach you how to select appropriate research projects).
What are legitimate methods and sources?
See a fourth year historiography text.
How does history adapt when there are no sources?
It ceases to be.
Perhaps the most important question: What are some questions that can never be answered by historical sources and methods?
Depends on how you feel about a number of major humanities problems. Can empirical work produce truth? Can you read texts with veracity? Importantly for the second sub-question: How? Do you believe that it is possible to recreate hypothesised internal states in subjectivities through correct reading of the records of the subject's personality (psycho-history)? Is it possible to accurately depict either in concrete or abstract the actual social forces of history via a correct reading of the records of the past (Thompson/Althusser debate)? Do collective subjectivities exist, and if so are they the only body capable of a real knowledge of the actual process of capitalism (praxis/theory debate in Marxism)? Is there a teleos?
History, like other empirical engagements with material reality, is the least worst possible way of uncovering conditional and limited knowledge about the world. While it is dependent upon reading texts to produce meaning, there are a number of ways out of the quagmire of post-modernism/post-structuralism other than a constant quagmire of interdetermination.