What actions does the Eruv Tavshilin permit one to do on behalf of Shabbos?^[1]

One who performs Eruv Tavshilin is permitted to perform all food related Melachos on Friday for Shabbos. However, it is forbidden for one to perform perform any action that does not relate to the Shabbos meal even if one performed Eruv Tavshilin.

The permitted actions: As stated above, it is permitted to do all actions that relate to the meals and foods of Shabbos. This includes the following actions:

1. Cooking: It is permitted to cook foods on Friday on behalf of Shabbos if the food will be ready before Shabbos.
2. Preparing food: It is permitted to prepare all foods on behalf of Shabbos even if it entails a Melacha that is generally forbidden on Shabbos [but permitted on Yom Tov] such as kneading, mashing, peeling, and the like. Hence, one may use a peeler on Friday to peel vegetables for the Shabbos meal, and may make fresh Tehina for the Shabbos meal, cook and then mash potatoes etc. etc.
3. Setting the table: It is permitted to set the table on Friday on behalf of the Friday night meal.
4. Dishes:^[2] It is permitted to wash dishes on Friday on behalf of the Shabbos meals.
5. Lighting candles: It is permitted to light a fire on Friday on behalf of having warmth or light on Shabbos. Thus, one who performed Eruv Tavshilin may light candles on Friday on behalf of Shabbos. It is however forbidden to light a candle on Friday on behalf of of a Yartzite or Yizkor. This applies even if one is not doing so on behalf of Shabbos but rather on behalf of Yom Tov on Friday.

The forbidden actions: As stated above, it is forbidden to perform non-food related Melachos for Shabbos. This includes the following actions which are forbidden:

1. Rolling the Torah scroll to the Shabbos reading:^[3] One may not roll the Sefer Torah on Friday to the Shabbos Torah reading.
2. Eruv Chatzeiros:^[4] One may not perform Eruv Chatzeiros on Friday on behalf of Shabbos.
3. Folding and preparing clothing and making bed:^[5] One may not fold and prepare clothing or make his bed on Friday on behalf of Shabbos.
4. Bathing and immersing in a Mikveh:^[6] It is permitted for a person to immerse in a mikvah on Friday on behalf of Shabbos. This applies even if he did not perform Eruv Tavshilin.^[7]

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[1] Admur 503:3; 393:1; Michaber 528:2; M”A 528:2; Levush 528; Kaf Hachaim 528:8; See Piskeiy Teshuvos 528:3; Nitei Gavriel Yom Tov 88; Siddur Eruv tavhsilin [Ades] p. 25-31; Vayaan Eliyahu [Shmuel] 10

[2] Admur 503:3; See Ohalei Shem 8:151

[3] Admur 503:3; M”A 503:1; 667:3; Maharil Tefilas Yom Tov p. 188; Machazik Bracha 667:2; Shaareiy Teshuvah 667; Chayeh Adam 153:6; M”B 667:5

[4] Admur 393:1; 503:3; Michaber 528:2

[5] Beir Yaakov 528:2 in opinion of Admur ibid; Chelkas Yaakov 3:84; See Nitei Gavriel ibid footnote 10

Other opinions: Some Poskim permit a person to fold clothing on Friday on behalf of Shabbos. [Elya  Raba 302:8; Hagahos Rav Akiva Eiger 302; M”B 302:6; Daas Torah 302; See Nitei Gavriel 88:4 footnote 10]

[6] Nitei Gavriel Yom Tov 88:15; Eruv Tavshilin Haruch 2:11; Beitza 17a [permitted]; Hemshech Samech Vav end of page 387 [p. 511 in new edition]; Erev Pesach Shechal Beshabbos 12 footnote 54 that from Beitza 17b it is proven that there is no issue with doing so and that indeed some are accustomed to do so; See Nitei Gavriel Yom Tov 56:7-8 footnotes 13-14 for different customs and opinions; Piskeiy Teshuvos 290:4 and footnote 26 and 444:12 and footnote 85 and 528:3 [permitted, and some do so, but don’t say it

Other opinions: Imrei Pinchas 799 [not to immerse]; Divrei Yechezkal Hachadash in name of the Tzadik of Shinava that one should not do so due to it being a belittlement; Kaneh Bosem 3:22-24 is stringent; Rav Moshe Freund in Ateres Yehoshua p. 93 127-128 and Kisvei Kodesh Ateres Yehoshua  Michtav 21 and Mara Deshmaasisa 184 [that personally he and his father would not immerse in such a case as Shabbos does not cause impurity, however, those who want to do so may do so and there is no Halachic issue involved]

[7] The reason: The act of immersion does not transgress the prohibition to prepare on Yom Tov for after Yom Tov, being that it is an action which one benefits from already on Yom Tov and appears simply like washing dirt off the body and cooling off in a body of water, and does not look like a preparation activity for after Shabbos. In fact, the Poskim state that it is for this reason that it is at all permitted for one to immerse in a Mikveh on Shabbos and is why doing so does not contain the prohibition of Tikkun Keli, as to the onlooker it simply seems like he is bathing the body. In essence, entering the Mikveh for after Shabbos purposes, is no different than one who sleeps or eats on Shabbos in order so he have strength after Shabbos, which is permitted to be done. This is explicitly ruled in the Mishneh Beitza which states that one who is impure is to immerse on Shabbos afternoon for the sake of Yom Tov. Nonetheless, some Poskim write that explicitly saying the purpose of the sleep, and that it is for the sake of after Shabbos is considered a belittlement of Shabbos and the same would seemingly apply to one who says the purpose of his immersion. However, it is clear from other Poskim, including Admur, that they are not worried of this issue, and that it is permitted to even make this statement. So can also be proven from the teaching of the Rebbe Rashab [brought below] that one should immerse in a mikvah on Yom Tov afternoon for the sake of Shabbos, and that there is no Halachic issue with preparing on Yom Tov for after Yom Tov regarding this matter.