[The Squatter of Amon Rûdh]

But did the choice follow automatically on being Half-Elven?

I think that this depends on one's definition of 'half-elven'. In Letter #345, Tolkien wrote: "Arwen was not an elf, but one of the half-elven, who abandoned her elvish rights." but he does not qualify this description. Obviously there is more to being 'half-elven' in this context than simply having one elven and one human parent, and the following passage from the Silmarillion explains matters a bit more clearly.

It is told among the Elves that after Eärendil had departed, seeking Elwing his wife, Mandos spoke concerning the his fate; and he said: 'Shall mortal Man step living upon the undying lands, and yet live?'
But Ulmo said: 'For this he was born into the world. And say unto me: whether is he Eärendil Tuor's son of the line of Hador, or the son of Idril, Turgon's daughter, of the Elven-house of Finwë?' And Mandos answered: 'Equally the Noldor, who went wilfully into exile, may not return hither.'
But when all was spoken, Manwë gave judgement, and he said: 'In this matter the power of doom is given to me. The peril that he ventured for love of the Two Kindreds shall not fall upon Eärendil, nor shall it fall upon Elwing his wife, who entered into peril for love of him; but they shall not walk again ever among Elves or Men in the Outer Lands. And this is my decree concerning them: to Eärendil and to Elwing, and to their sons, shall be given leave each to choose freely to which kindred their fates shall be joined, and under which kindred they shall be judged.'

Now, this passage still leaves us with a problem concerning Arwen: clearly those descendents of Eärendil and Elwing to whom Mandos is giving a choice are Elrond and Elros. No mention is made here of their descendents.

Fortunately Tolkien has resolved this issue as well. In Appendix A to The Lord of the Rings, he clarifies the whole issue. Having explained that "The sons of Eärendil were Elros and Elrond, the Peredhil or half-elven." he tells us:

At the end of the First Age the Valar gave to the Half-elven an irrevocable choice to which kindred they would belong. Elrond chose to be of Elven-kind, and became a master of wisdom. To him therefore was granted the same grace as to those of the High Elves that still lingered in Middle-earth: that when weary at last of the mortal lands they could take ship from the Grey Havens and pass into the uttermost West; and this grace continued after the change of the world. But to the children of Elrond a choice was also appointed: to pass with him from the circles of the world; or if they remained to become mortal and die in Middle-earth.

Therefore, the choice of kindreds is offered only to the Half-elven, those being Elrond and Elros. The term extends to cover Elrond's children, but with a modification in the terms of the choice: their decision is whether or not to accompany their father, the choice of kindred following from the path taken. Presumably this privilege is granted to Elrond's children for two reasons: firstly because their father's choice of the fate of the Elves pre-dated their births; and secondly because they were descended from three senior Eldarin houses: from Olwë through Finarfin, Galadriel and Celebrían; from Elwë through Lúthien and Elwing, and from Finwë through Fingolfin and Idril.

My apologies to Sharkû, Legolas and Saucepan, who knew all of this already; but this is the Novices and Newcomers forum, so I thought I'd better clarify my terms a little. Obviously where I said 'half-elven' above I meant it in the sense that Tolkien meant it in his letter, not as a general term for those with parents of either kindred.