While mideoraisa one may cook on Yom Tov for Shabbos (Pesachim 48b), Chazal were concerned that people may take it a step further and cook for a weekday meal. Eruv tavshilin ensures that the Shabbos preparations begin before the onset of Yom Tov, and any preparations done on Yom Tov itself, are simply a continuation of that. It also serves as a reminder not to do any other preparations other than for Shabbos (Beitza 15b, Rashi, and Mishna Berura 527:25). The Shulchan Aruch (OC 527:13) writes that when Yom Tov is on Thursday and Friday, one must not begin preparations on Thursday (1st day Yom Tov) but rather must wait until Friday.
One should use a matza or bread along with a cooked food, such as eggs, meat or fish (Mishna Berura 527:11) which should ideally be cooked on erev Yom Tov (Biur Halacha, 527:14).
The Mishna Berura (527:48) writes that it is best to use this bread as the second bread of lechem mishna for both Friday night and Shabbos lunch and eat it during shalosh seudos. This way, one continues using an item used for one mitzvah for other mitzvos. The Mishna Berura (527:4) writes that one who forgot to make his eruv before shekia may do so (even with a beracha) during bein hashemashos (the period of time between shekia and tzeis hakochavim, nightfall). This wouldn’t apply once the Shul had begun davening maariv or he had otherwise accepted Yom Tov upon himself. As there is a machlokes as to whether one needs to make an Eruv tavshilin to light candles, R’ Mordechai Karmy (Ma'amar Mordechai 527:18) paskens that one who isn’t cooking at all, should do so without a beracha in order to enable them to light candles and warm up food, etc.
One who doesn't understand the Aramaic declaration must say the English translation (Mishna Berura 527:40).
A husband who makes an eruv for his family should ideally do so in front of his wife (Aruch Hashulchan 527:22), and should intend to include his family. Any guests (or others) who wish to be included should make a kinyan to share ownership of the food. (Someone else can do this on their behalf. See Shulchan Aruch OC 527:10)
While the Rav makes an eruv on behalf of anyone in his community who forgets or loses theirs, one can’t rely on this instead of choosing to make one. One who forgets two consecutive times is no longer considered accidental and it would not help to rely on this (Baer Hetev 527:6).
If one arrived in Shul and realized that they had forgotten to make an Eruv before Yom Tov, they should go home if they can still make one before Yom Tov. Alternately, one may call home and ask someone else (such as one’s wife) to do so. If not, there are some (Tiferes Yisroel, Beitza 2:1) who hold that one can designate food that he has at home. He can’t make the beracha, and must omit the words behadein eruva, with this eruv. R’ Yitzchak Yaakov Weiss (Minchas Yitzchak 7:36) held that one can’t rely on this.
The Mishna Berura (527:4) writes that one who forgot to make his eruv before shekia may do so (even with a beracha) during bein hashemashos (the period of time between shekia and tzeis hakochavim, nightfall). This wouldn’t apply once the shul had begun davening maariv or he had otherwise accepted Yom Tov upon himself.In an emergency, the Shulchan Aruch (OC 527:22) even allows making one under strict conditions on the first day Yom Tov (in chutz la’aretz), though this doesn’t apply on Rosh Hashana (as there’s no safeik deyoma).