When the (first) Seder falls out on Friday night (as it did this year), the custom at my family's Seder is not to say Shalom Aleichem and Eishet Chayil. I have looked in all of the Machzorim and Haggadot in the house, and none of them has Shalom Aleichem and Eishet Chayil, so I assume the prevalent custom is not to say it. However, I have no idea if my assumption is correct, and I also have not been able to find any sources to indicate any reasons why Shalom Aleichem and/or Eishet Chayil are or are not recited on the night of the Seder. Does anyone know of any sources for saying or not saying them on the night of the Seder? Is it specific to the Seder, or is it a general rule for any Yom Tov that falls out on Shabbat?
The Piskei Teshuvos Siman 473:2 writes that even those who are noheg to say sahlom aliechem and eishes cheil on Yom tov should not say it the night of pesach in order to hurry and make kiddush right away,since it is a mitzva to do so.(Vayged Moshe 13:2)
However,he brings in footnote 9 that there are those who still say these items such as Chassidei Viznitz,and the Belzer Rebbe. He also brings those who say them quietly(Minhagim at the end of Shulchan Aruch HaRav)
From the Oz V'hadar Mesivta Haggadah page 68:
There is a dispute whether one should say Zemiros Shabbos on Yom Tov. The Matteh Ephraim (583:1) says that one should say Shalom Aleichem on a Shabbos Rosh HaShanah. Likewise, the Kaf HaChaim (583:3) writes likewise, adding that it is true for all yamim tovim.
The Elef HaMagen (ad. loc.) argues, saying that one should not sing these zemiros on Shabbos Yom Tov, and not even on Shabbos Chol HaMoed. Likewise, the Divrei Yetziv (OC 123:1-2) writes that the Divrei Chaim would not say Shalom Aleichem on Shabbos Yom Tov, as it's written for a Shabbos seudah, not a Yom Tov seudah.
In the middle ground is the Minhagei Kol Aryeh and Darchei Chaim v'Shalom (514), who would not sing Shalom Aleichem for the same reason as the Divrei Yetziv, but they would still sing Eishes Chayil.
There's also the opinion of the Rov Po'alim (1:13) who says that although it is fitting to say them (because one normally says them on Yom Tov, like the first set of opinions above), it is our custom not to say them. Some (that's what the Mesivta says - "yeish shekasvu") explain that since we're trying to keep the kids interested and not have them fall asleep early, we don't want to drag things out. Nevertheless, many say zemiros anyway. (It should be noted that they are discussing the Seder night only. According to this opinion, Shabbos day zemiros should still be recited.)