Collection of Questions on the Laws of Eruv Tavshilin

Shalom to the honored Rabbi,
 
 I have several questions regarding Eruv Tavshilin:
 
 A. When Yom Tov falls on Thursday and Friday (in the Diaspora, where there are two days of Yom Tov), and Shabbat comes immediately afterwards — does the Eruv already permit, on the first day of Yom Tov (i.e., Thursday), cooking and preparing food directly for Shabbat, or is the leniency only applicable on the second day of Yom Tov, which is Erev Shabbat?
 
 B. In the text of Eruv Tavshilin we say:
 “בהדין עירובא יהא שרא לנא לאפויי ולבשולי ולאטמוני ולאדלוקי שרגא ולמעבד כל צרכנא מיום טוב לשבת”.
 
 Regarding “ולאדלוקי שרגא” — does this refer to the actual lighting of the Shabbat candles, and is an Eruv Tavshilin required for this as well? Or does it perhaps refer only to lighting a fire for the purposes of cooking and baking?
 
 C. Is an Eruv Tavshilin also required for preparations that do not involve actual cooking work, such as setting the table, making the beds, and other forms of preparation for Shabbat, because of the prohibition of preparing (hachanah — preparing from Yom Tov for Shabbat), or is the Eruv needed only for labors such as baking, cooking, and the like?
 
 D. Can a woman perform the Eruv Tavshilin herself and recite the blessing over it, or is it preferable that the husband make the Eruv and the blessing? And if both can — is there any halachic preference as to which of them should do it?
 
 E. In a case where all the food for Yom Tov and Shabbat was already prepared on Erev Yom Tov, and one does not intend on Yom Tov itself to cook, bake, or do cooking‑related labors for the sake of Shabbat — is there still an obligation to make an Eruv Tavshilin?
 
 For example, if in practice the only actions that will be done on Yom Tov for the sake of Shabbat are lighting the Shabbat candles, setting the table, washing dishes, taking food out of the freezer, cutting vegetables, and other forms of preparation only — but without cooking, baking, or lighting a fire for cooking — is there a need to make an Eruv Tavshilin? And if so, is it done with a blessing or without a blessing?
 
 If you could also cite the halachic sources for these matters, I would greatly appreciate it.
 
 Thank you very much.

Shalom and blessings, 

A. The Eruv permits cooking only on Friday. (Mishnah Berurah, siman 527, se’if katan 41). 

B. The intention is the lighting of the candles. The poskim disagree whether an Eruv is required for this (see Shulchan Aruch, Orach Chaim 527:19, and Mishnah Berurah there). Practically, the Mishnah Berurah writes that it should be mentioned in the text of the Eruv, since the primary view follows those who are stringent in this matter. 

However, since there are dissenting opinions, one who knows that he will not cook and the like, but will only light the candles, should make an Eruv but not recite the blessing over it. (Ruling of our teacher, Rabbi Amram Fried shlita).

C. On this there is a dispute. The Mishnah Berurah (siman 307, se’if katan 17) writes: "ביום טוב שחל בע"ש מותר לקפל טליתו כשפושטו אם היא חדשה ולבנה אף שאין דעתו ללבשו בו ביום והיינו כשעשה עירובי תבשילין דאל"ה הרי אסור לו להכין לצורך מחר כ"כ בא"ר אבל בחידושי רע"א הוכיח דאף בלא הניח ע"ת שרי אם היא סמוכה לשבת" (On a Yom Tov that falls on Erev Shabbat, it is permitted to fold one’s tallit when taking it off if it is new and white, even though he does not intend to wear it again that day, and this is when he made an Eruv Tavshilin; otherwise, it is forbidden for him to prepare for the needs of tomorrow — so wrote the Ba’er Heitev. But in Chiddushei Rabbi Akiva Eiger he proved that even if he did not make an Eruv Tavshilin, this is permitted if it is close to Shabbat)."
On the other hand, the Mishnah Berurah (siman 528, se’if katan 3) writes: "דעירובי תבשילין אינו מתיר אלא לתקן צרכי סעודה לצורך מחר" (Know that an Eruv Tavshilin only permits arranging/preparing matters related to the meal for the sake of the next day)."
In practice, our teacher Rabbi Amram Fried shlita writes: “If one set aside an Eruv Tavshilin, then according to the strict law it is permitted to tidy up the house at a time when there is still enough of the day left to benefit from it (as mentioned above in se’if 100). And it is better to tidy the house while it is still broad daylight, so that it not be noticeable that he is doing this specifically in honor of Shabbat.” (and he cites Mishnah Berurah 503:1 and Sha’ar HaTziyyun 503:2). 

D. A woman may perform an Eruv Tavshilin, and when doing it for herself she can thereby acquire (include) others in it as well. (Mishnah Berurah 527:34). 

E. In all such cases one should make an Eruv Tavshilin without a blessing. (As explained in the sources cited above).