I thought they did not eat with him, but I may be mixing up the Qur'an's version of the story:

051.024
Has there come to you information about the honored guests of Abraham?

051.025
Behold, they entered his presence, and said: "Peace!" He said, "Peace!" (and thought, "These seem) unusual people."

051.026
Then he turned quickly to his household, brought out a fatted calf,


051.027
And placed it before them.. he said, "Will ye not eat?"

051.028
(When they did not eat), He conceived a fear of them. They said, "Fear not," and they gave him glad tidings of a son endowed with knowledge.

051.029
Then his wife came up in great grief, and she struck her face and said: An old barren woman!

051.030
They said, "Even so has thy Lord spoken: and He is full of Wisdom and Knowledge."

051.031
(Abraham) said: "And what, O ye Messengers, is your errand (now)?"

051.032
They said, "We have been sent to a people who are (deep) in sin.

051.033
"That we may send upon them stones of clay,"

051.034
"Marked as from thy Lord for those who trespass beyond bounds."

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As for the Biblical version, one website states that Rabbi Baruch S. Davidson says "What about the angels? How could they eat non-kosher? According to one opinion, the angels didn't eat at all; they merely appeared to be eating, out of respect for their host."

In the Bible, Jesus seems to imply that spirit beings do not eat food:

Luke 24:36-46

[36] While they were still talking about this, Jesus himself stood among them and said to them, "Peace be with you."

[37] They were startled and frightened, thinking they saw a ghost. [38] He said to them, "Why are you troubled, and why do doubts rise in your minds? [39] Look at my hands and my feet. It is I myself! Touch me and see; a ghost does not have flesh and bones, as you see I have."

[40] When he had said this, he showed them his hands and feet. [41] And while they still did not believe it because of joy and amazement, he asked them, "Do you have anything here to eat?" [42] They gave him a piece of broiled fish, [43] and he took it and ate it in their presence.

[44] He said to them, "This is what I told you while I was still with you: Everything must be fulfilled that is written about me in the Law of Moses, the Prophets and the Psalms."

[45] Then he opened their minds so they could understand the Scriptures. [46] He told them, "This is what is written: The Christ will suffer and rise from the dead on the third day."

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One could say that spirit beings who take human form go so far as to be able to eat food, but I don't know if it is clear how material they are, or just appearing in human forms.
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Sorry if I used bad examples. I personally don't believe that angels mated with human beings though. Other groups of people or individuals have had terms that were thought to apply to these angels specifically (sons of God) and it didn't mean those people were angelic beings.

In any case, angel comes from a greek word which means messenger due to the hebrew word for angel meaning messenger.

From Wikipedia:

"A Mal'akh also: Mal'ach or Mal'aqh; plural Mal'akhim) is a messenger of God, an angelic envoy or an angel in general who appears throughout the Hebrew Bible, Rabbinic literature, and traditional Jewish liturgy. "

"Hebrew "Mal'akh" from the Semitic consonantal root l-'-k meaning "to send". This root is attested in Hebrew only in this noun and in the noun "Melakha" meaning "work"."

I think you have the right idea about these things! I think people twist scripture alot and end up getting entirely distracted from its central message which is to worship and obey God.