Why did Elrond choose Legolas as representative of the Elves and Gimli as representative of the dwarves?

Question

Beside Frodo who volunteered to be the Ring bearer and his Hobbits friends who volunteered as well, the rest of the Fellowship members were named or chosen by Elrond:

‘And I will choose you companions to go with you, as far as they will or fortune allows. The number must be few, since your hope is in speed and secrecy. Had I a host of Elves in armour of the Elder Days, it would avail little, save to arouse the power of Mordor.

‘The Company of the Ring shall be Nine; and the Nine Walkers shall be set against the Nine Riders that are evil. With you and your faithful servant, Gandalf will go; for this shall be his great task, and maybe the end of his labours.

‘For the rest, they shall represent the other Free Peoples of the World: Elves, Dwarves, and Men. Legolas shall be for the Elves; and Gimli son of Glóin for the Dwarves. They are willing to go at least to the passes of the Mountains, and maybe beyond. For men you shall have Aragorn son of Arathorn, for the Ring of Isildur concerns him closely.’

The Lord of the Rings Book Two, Chapter 3: The Ring Goes South Page 275-6 (Single volume 50th Anniversary Edition)

The question is why Legolas and Gimli were chosen and not other representatives of their races.

Possible answers:

1. Gimli and Gloin were the only dwarves in Rivendell at that time, and Gloin was too old for the quest.
2. Gimli and Legolas were among the most formidable warriors of their races, and thus Elrond has chosen the best fighters to escort and protect Frodo.

Answer 1

Because they were headed that way anyways

The answer is in the line immediately following the one you had bolded.

Legolas shall be for the Elves; and Gimli son of Glóin for the Dwarves. They are willing to go at least to the passes of the Mountains, and maybe beyond.

It was never the plan for the fellowship to accompany Frodo the whole way. Many of them had different destinations in mind (Aragorn and Boromir were headed to Gondor, Legolas to Mirkwood, and Gimli to Erebor.). But they would all at least need to cross the Misty Mountains together, and so set off in a group.

If you look a few lines past where you quoted, you'll see that Aragorn and Boromir are also explicitly said to only be in the group for convenience.

‘I would have begged you to come,’ said Frodo, ‘only I thought you were going to Minas Tirith with Boromir.’
‘I am,’ said Aragorn. ‘And the Sword-that-was-Broken shall be re-forged ere I set out to war. But your road and our road lie together for many hundreds of miles. Therefore Boromir will also be in the Company. He is a valiant man.’

And as Elrond reiterates right before they set off, everyone else was free to turn aside to their other paths as needed.

The others go with him as free companions, to help him on his way. You may tarry, or come back, or turn aside into other paths, as chance allows.

Answer 2

The reasons were different in the two cases.

For Gimli, it's very simple: There were only two dwarves in Rivendell and Gimli was the younger (and probably the more fit.)

Introducing Gimli's father:

Next to Frodo on his right sat a dwarf of important appearance, richly dressed. His beard, very long and forked, was white, nearly as white as the snow-white cloth of his garments. He wore a silver belt, and round his neck hung a chain of silver and diamonds.

...Not a person to be sent out on a few months of roughing it.

For Legolas, it mainly goes back to the complicated splits among Elves. The High Elves (who were the main Elves in Rivendell and I think the only Rivendell Elves named in LotR) were a small minority of the Elves, forming an aristocracy. Gandalf and Elrond had already concluded that the inherent power of these great Eldar would not help,

'Even if you chose for us an elf-lord, such as Glorfindel, he could not storm the Dark Tower, nor open the road to the Fire by the power that is in him.'

so selecting as the representative of the Elves a Sindarin Elf was definitely politic and, at worst, did no harm.

Once a Sindarin Elf was decided upon, Legolas was an obvious choice. He was a prince of the nearest (and probably biggest) Sindarin Elvish community in the western Middle Earth. (Doubtless there were Sindarin Elves living in Rivendell -- The Hobbit suggests this -- but any such person would be viewed by other Elves as Elrond's representative, not as representative of Elves as a whole.)

Elvish-politically, the representative had to be a Sindarin and had to be from outside the Rivendell community. Hence, Legolas was a great choice.

Additionally, they were both present at the Council of Elrond, and it made sense to send an Elf who had been privy to the discussions.

Note that both Legolas and Gimli were not just present at the Council but were destined to be at there. Their presence was providential. I doubt that either Elrond or Gandalf would ignore hints given them by Eru.

'That is the purpose for which you are called hither. Called, I say. though I have not called you to me, strangers from distant lands. You have come and are here met, in this very nick of time, by chance as it may seem. Yet it is not so. Believe rather that it is so ordered that we, who sit here, and none others, must now find counsel for the peril of the world.

Their selection was for many reasons not by chance.