Who did what at the story of the destruction of S'dom and Amorah

Question

I would like to ask two questions with regards to the context I applied with these questions:

1. Who is who?
2. What happened to the third men/malach?
3. Who did destroy S'dom and Amorah?

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Genesis 18:1 tells us that HaShem appeared unto Avraham in the terebinth trees of Mamre.

And then the story continues: Genesis 18:2 And Avraham lifted up his eyes and looked, and, hinei, three man (shloshah anashim) stood over against him; and when he saw them, he ran to meet them from the tent door, and bowed down to the earth and said: 'My lord (singular), if now I have found favour in thy sight (singular), pass not away, I pray thee, from thy servant.

After some lovely meal the story goes on: Genesis 18:16 And the men rose up from thence, and looked out toward Sodom; and Abraham went with them to bring them on the way.

On this point in time HaShem is wondering wether He should inform Avraham of His plans with S'dom and Amorah, i.e. that He would go down and see whether they have done altogether according to the crying out [for punishment] of it, which has come unto Him, and if not HaShem would know.

And then this is what happens: Genesis 18:22 And the men turned their faces from thence, and went toward S’dom; but Avraham stood yet before HaShem. And after a conversation with HaShem the text reads: Genesis 18:33 And HaShem went His way, as soon as He had left communing with Avraham; and Avraham returned unto his place.

The story picks up at Genesis 19:1 And two malachim came to S'dom, and Lot sat in the gate of Sodom; and Lot saw them, and rose up to meet them; and he fell down on his face to the earth (quite similar reaction as Avraham had). and he said: "Behold now, my lords (plural), turn aside, I pray you, into your servant's house.."

In Genesis 19:12 (and other verses before) these malachim are described as men and they tell Lot to take his family and bring them out of this place. (verse 13:) for they will destroy this place, because the cry of them is waxed great before the HaShem; and HaShem hath sent us to destroy it.' Genesis 19:14 And Lot went out, and spoke and said: 'Get up and go out of this place; for HaShem will destroy the city.'

Genesis 19:15 And when the morning arose, then the malachim (heavenly messengers) hastened Lot, saying: 'Arise, take thy wife, and thy two daughters that are here; lest thou be swept away in the iniquity of the city.But he lingered; and the men laid hold upon his hand, and upon the hand of his wife, and upon the hand of his two daughters; the LORD being merciful unto him. And they brought him forth, and set him without the city.

Lot prays to HaShem and is heard.

Genesis 19:24 Then HaShem caused to rain upon Sodom and upon Gomorrah brimstone and fire from HaShem out of heaven

Genesis 19:27 And Abraham got up early in the morning to the place where he had stood before HaShem. And he looked out toward S'dom and Amorah, and toward all the land of the plain, and beheld, and, lo, the smoke of the land went up as the smoke of a furnace. And it came to pass, when G-d destroyed the cities of the plain, that G-d remembered Abraham (plead with Him), and sent Lot out of the midst of the overthrow (because of his rightousness?), when He overthrew the cities in which Lot dwelt.

Answer 1

The answer to your question, “What happened to the third men/malach?” is I suggest answered by Rashi to 18 (33). The possuk says,

And the Lord departed when He finished speaking to Abraham, and Abraham returned to his place.

Rashi says,

(And the Lord departed, etc.: Since the defender was silent, the Judge left.)

and Abraham returned to his place: The Judge left, the defender left, and the prosecutor is accusing. Therefore: “And the two angels came to Sodom,” to destroy (Gen. Rabbah 49:14). One to destroy Sodom and one to save Lot, and he [the latter] is the same one who came to heal Abraham,

but the third one, who came to announce [Isaac’s birth] to Sarah, since he had performed his mission, he departed

(Tan. Vayera 8).

The answer to your question “Who did destroy S'dom and Amorah?” is Hashem. It's clearly stated in 19 (24 and 27) which you quote. But to understand it better look at the comment of Rashi to 19 (22). The Malach who thought he would destroy the cities says to Lot,

“Hasten, flee there, for I will not be able to do anything until you arrive there." Therefore, he named the city Zoar.”

on which Rashi says,

Hasten, flee there, for I will not be able to do: This is the angels’ punishment because they said (verse 13): “For we are destroying,” and they attributed the matter to themselves; therefore they did not move from there until they were compelled to say that the matter was not in their power. — [based on Gen. Rabbah 50:9]

and Rashi clarifies the roles of the malachim saying:

for I will not be able: [This expression is] in the singular. From here you learn that one was to overturn [the cities] and one was to save [Lot and his family], because two angels are not sent for one mission. — [from Gen. Rabbah 50:2]

Your first question "Who is who" is at least partailly answered from the above.