My question is this: The view expressed on this page by some of the rabbis was that one should not eat with a goy on Yom Tov. How is this resolved against the fact that Avraham Avinu opened up his tents to serve the Melachim, who as far as he knew, were goyim? I understand that they even came on Pesach.
Nice question. According to the Midrash (cited by Tosfos Rosh Hashanah 11a DH Ela, and Da'as Zekeinim Bereishis 18:14 -- see Rashi Bereishis 19:3), the angels came, looking like Arab nomads, on the 15th of Nisan, the first day of Pesach. How could Avraham cook for them?
The Sefer "Pardes Yosef" discusses how Avraham was permitted to cook for non-Jews on Yom Tov (Bereishis 18:8), but he only relates to the Isur d'Oraisa, and not to the Isur d'Rabanan mentioned in our Gemara. We know that Avraham kept even the Rabbinic prohibitions, so why wasn't he careful not to have such guests, "Shema Yarbeh Bishvilam?" Here are some possible answers:
(a) A similar question arises in the commentators on the Shulchan Aruch (YD 101). The Halachah is that a piece of food that is normally "fit to be served to a guest to honor him" ("Chatichah ha'Re'uyah l'Hiskabed") which became prohibited (for example, it came from an animal that was not properly slaughtered), will not become Batel when it falls into a mixture, even though the quantity of permitted food is much greater than that of the prohibited food. The commentators give examples of food which is "Re'uyah l'Hiskabed." One example is cow's tongue, as we learn from Avraham Avinu who served that to his guests (TAZ, YD 101:8).
However, there is another Halachah which states that we do not determine what food is "Re'uyah l'Hiskabed" by what is served to non-Jews, but only by what is served to Jews (PISCHEI TESHUVAH, YD 101:4, citing the NODA B'YEHUDAH). If so, how can we learn from Avraham Avinu that tongue is a "Chatichah ha'Re'uyah l'Hiskabed," if he was serving it to non-Jews?!
The most straightforward answer is that before the Torah was given, and the Jewish people had not yet taken full form, what was considered a "Chatichah ha'Re'uyah l'Hiskabed" could not be determined by what was normally given to members of the Jewish people, because there was no Jewish people, and therefore it was determined by what was given to any guest. Similarly, regarding your question, the prohibition against cooking on Yom Tov for non-Jews was not possible to observe (at least in the form that it was possible to observe later), since there was not yet an entity called the Jewish people. (Y. Shaw)
This, however, touches upon the famous discussion of the Mishneh la'Melech (in Perashas Derachim, ch. 1), as to whether or not our forefathers had the status of "Jewish People", even though the Torah was not yet given. If they did not have such a status, it is clear that there was no reason for them to distinguish between other members of society and themselves, as far as cooking on Yom Tov. But what about those who say the forefathers did have the status of "Jewish People" back then?
(b) For one, the Gemara does not say, "One should not eat with non-Jews." It only prohibits inviting them to eat, lest one prepare food exclusively for them. In Avraham's case, nomadic wanderers stumbled by his tent on the hottest day of the year. It is not inconceivable that such a situation would be bordering on Piku'ach Nefesh -- there lives would be endangered if they did not get fed and given to drink. If the non-Jew's life is in danger, it is probably permitted to feed him (if the food is not cooked in a separate pot).
(c) B'Derech Drush, it may be suggested that Avraham thrived on Hachnasas Orchim to such an extent, that he would have no Simchas Yom Tov without it. For him, then, cooking for the non-Jewish guests would be Melachah done for himself , and not for a non-Jew! (This is somewhat similar to the logic of the Gemara on 20b, "it is not proper for your table to be full and your master's to be empty.") Therefore there was no problem for him to cook for his guests, even in separate pots.
In addition to the answers which you offered in your response, it could be,also, that since Avraham Avinu kept the Mitzvos in the form of "Eino Metzuveh v'Oseh" (see Shitah Mekubetzes to Bechoros 6b), and according to the Ran in Sanhedrin, Bnei Noach have a Chiyuv d'Oraisa of Tzedakah, the Mitzvah of Tzedakah that Avraham Avinu was obligated to do (to feed the hungry wayfarers) is Docheh the Isur of cooking for Nochrim on Yom Tov, since that Isur he kept only as an Eino Metzuveh v'Oseh.