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 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.

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, it had to be a Sindarin and it had to be from outside the Rivendell community. Hence, Legolas as a great choice.

Finally, both Legolas and Gimli were destined to be at the Council and to take part in it. 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.

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.