"Question: What happened to the 'missing pieces' of the figures?"
An excellent question. A large number of the marbles had already been damaged, defaced, pillaged, reused as building material or ground down to make cement by the time Lord Elgin saw what remained and I imagine many of the pieces he took may have suffered further or met a similar fate if they had been left behind.
In 1816 Elgin noted: "Every traveller coming added to the general defacement of the statuary in his reach: there are now in London pieces broken off within our day. And the Turks have been continually defacing the heads; and in some instances they have actually acknowledged to me that they have pounded down the statues to convert them into mortar." (St Clair, Lord Elgin and the Marbles, OUP 1983, p.97)
That this was not a fabrication by Elgin is confirmed by earlier visitors. "In 1749 the traveller Dalton drew twelve figures in the west pediment of the Parthenon: by the time Lusieri arrived in 1800 there were only four. Five slabs of the frieze drawn by Stuart between 1750 and 1755 had completely disappeared. One slab of which a mould was taken by Fauvel as recently as 1790 was utterly destroyed. The metopes tell a similar story." (ibid.)
Let's be under no illusion. Elgin did not deface an intact or stable monument; by the time he arrived, the Parthenon was already disappearing at an alarming rate. Evidence indicates that neither the Turks nor most of the local Greeks were particularly bothered at the time (it was not until much later that the Greeks adopted the Parthenon as a national monument). It was a frantic race between the British and the French to grab what was left and save it from further damage; if the British had not taken the sculptures, the French would have. The British won. But there is much to suggest that the rivalry was more in the nature of who got the best trophy than who was more conscientious in matters of conservation.
The legality of Elgin's actions was contentious. He had a firman (letter of permission) issued in 1801 by the Turkish authorities but it was somewhat ambiguous and a letter he wrote to Spencer Perceval, the then Prime Minister, in 1811 suggests that Elgin was aware that he may have been going beyond its remit: "I had no advantage from the Turkish government beyond the Firman given equally to other English travellers. My successors in the Embassy could not obtain permission for the removal of what I had not myself taken away. And on Mr Adair's being officially instructed to apply in my favour, he understood, 'The Porte denied that the persons who had sold those marbles to me had any right to dispose of them'." (Hitchens, The Elgin Marbles: Should they be returned to Greece?, Verso 1997, p.38)
The reaction when the sculptures arrived in London was mixed. Byron famously condemned their removal from Athens as "Thy walls defaced, thy mouldering shrines removed by British hands"; Keats praised them as "Grecian grandeur".
Meanwhile, the debate continues as to whether the sculptures should stay in London or now be returned to Athens ...