Rashi explains based on the gemara (Baba Metziya 86) that although angels cannot eat, the angels who came to Avraham pretended to eat so as to not deviate from the normal practice of the community. The lesson for us is to blend in when blending in is appropriate. However, Tosfos on that sugya quotes the Midrash that contrary to the gemara's assumption, the angels did in fact eat. It is hard to say that there is a machlokes in "metziyus" whether angels can or cannot eat, so there must be some other explanation of what the underlying issue here is. I once tried to explain the gemara's position based on the view that the angels came on Pesach and matzah was served. The process of making matzah, turning raw wheat into a finished baked product, represents the sanctification of the physical world, a process angels are removed from and incapable of engaging in. Rav Shteinman in his Ayeles haShachar explains the two views as disagreeing over whether an angel can adopt a physical body for the purpose of fulfilling a mission or not. I'm not sure how much this adds to our understanding of the issue, as it begs the follow up question of why an angel can or cannot an angel appear in physical form -- in other words, the basis for that dispute -- which seems to bring us full circle back to where we started.
My wife's grandfather, R' Dov Yehudah Shochet, apparently (the return letter is published, but not the letter he sent) suggested in a different context that the machlokes Rishonim whether angels have a body or not (which I am not familiar with) can be explained as being no machlokes at all because it all depends on what the meaning of the word "body" is. Yes, angels have a body if you mean some type of spiritual envelope for their presence, but no, angels do not have a body if you mean a physical form. The Lubavitcher Rebbe replied to this letter with proof from the Rambam that no body means no body, i.e. no form whatsoever. Perhaps this machlokes is reflected in the differing views of the gemara and Midrash.
The Rishonim and Achronim discuss how Avraham could have served milk and meat to his guests when Chazal tell us that he obeyed the entire Torah even before it was given. The Da'as Zekeinim answers simply that the milk was served first followed by the meat. There are numerous other creative answers. Malbi"M writes that the cow served was no ordinary cow, but was an animal created through kabbalistic means using sefer yetzirah. Proof comes from the pasuk itself -- "es habakar asher asah," the cow which was made, not just prepared. The assumption is that this type of cow is not really meat and can be eaten together with milk. When discussing this at home the suggestion came up that these different views may help explain the difference between the gemara and Midrash (this is obviously speculative). If the meat served was physical beef, as the Da'as Zekeinim understood, then it is no wonder angels could not eat it, but if the beef was mystical beef created with the sefer yetzira, perhaps this type of food might be on the menu even of an angel.
On a final tangential note, the Malbi"m's assumption that meat created miraculously does not have the properties of what we call meat reminded me of a famous kashe of R' Chaim Brisker. The Beis Yosef asks why we celebrate Chanukah for eight days when there was enough oil for one day. One answer is that the jug of oil miraculously refilled itself each day, including the first, upon being emptied. R' Chaim questions why this miraculously created oil was acceptable for use in the menorah when the Torah demands that specifically olive oil be used. Miracle oil may look, taste, and feel like olive oil, but it did not come from olives! I saw another blogger discuss this recently and will leave you with the link and something to mull over as we get closer to Chanukah.