A. Almost certainly, yes, if by that you mean that he was capable of being in love with men. The love of his life seems to have been a male friend of his from Oxford, Moses Jackson, who was not a homosexual and may not even have been aware of this crush of Housman's. It's not clear whether Housman ever had sex with another man, or indeed with anyone at all. In those days people and biographies were more discreet, and Housman was a very private person, except in his poetry. `Because I liked you better' can be read in a new light if you bear this in mind. `Oh who is that young sinner' is known to be about the notorious Oscar Wilde trial.
A. Not in the sense you probably mean, despite the valiant attempts of some educators to claim this. A Shropshire Lad was published in 1896, and so has nothing to do with the First World War. Internal evidence in the `war poems' in Last Poems (1922) also suggests that they are not about that conflict, the more so as Housman's own introduction dates most of them to between 1895 and 1910. Of course, there were plenty of other colonial wars before then, and these, not WWI, provide the backdrop to Housman's writing.
A. In a letter in 1933 Housman stated that the main influences of which he was conscious were the songs of Shakespeare, the Scottish Border Ballads and the German poet Heinrich Heine. He specifically denied any strong influence from his background in classical literature (though many have claimed to see it). There are occasional clear classical references, such as the `strengthless dead' in `To an athlete dying young' (a Homeric phrase), the free translation of Horace's Diffugere Nives, or the poems on classical subjects, such as Atys, but these don't really constitute an influence on his style as a poet.
A. The Society of Authors in the UK is the current representative of the Housman estate. The poems are out of copyright in the UK now, so in my opinion you don't need to ask anybody. This is not legal advice, though, and it would be particularly unsafe to rely on it if you are outside the UK.
A. The King who reigned in the East was Mithridates VI, King of Pontus (in modern-day northern Turkey). He is supposed, according to legend, to have made himself immune to poisoning by taking small doses of various poisons throughout his life (`He gathered all that sprang to birth / From the many-venomed earth;'). He died by suicide at the age of 69, a respectable age for a monarch in those days, when the Romans conquered his kingdom; the story goes that he was unable to kill himself with poison and had instead to use a sword. The point of the verse is that if you innoculate yourself with small doses of unhappiness (in the form of Terence's verses), as Mithridates did with poisons, then you will be proof against your own real troubles when they come along (`if the smack is sour / The better for the embittered hour').
A. A Shropshire Lad was published in 1896, Last Poems in 1922. More Poems and Additional Poems were published posthumously by Housman's brother Lawrence, the former in 1936, the latter as part of his memoir of Housman, A.E.H., in 1937.
A. Stevenson wrote them first:
Under the wide and starry sky Dig the grave and let me lie: Glad did I live and gladly die, And I laid me down with a will. This be the verse you grave for me: Here he lies where he long'd to be; Home is the sailor, home from sea, And the hunter home from the hill.You can interpret `This be the verse that you grave for me' as `say this about me if you want a memorial' (for `grave', read `engrave') and so it was altogether suitable that Housman used the lines in his memorial to Stevenson, written after Stevenson's death.
A. `To an athlete dying young'. You can find this out the same way I did, by typing `Housman out of Africa' into Google.