The Shulchan Aruch (Remo in 271:12) indeed says that one should wash before Kiddush and make Hamotzi immediately after Kiddush- and the Yekkes (Jews whose ancestors lived in Germany) actually do so. See Mishna Brura Vol 3. pg 95-96 for details. Note that even according to those that say that one should wash after Kiddush, pasken that the Man of the House (the person making Kiddush) should do so; everybody else should wash before Kiddush. (ibid)

At a kosher meal in the home of an observant Jewish family, every meal begins with

the blessing "hamotzi", expressing gratitude for food to meet our needs and recited

over bread. No additional blessings are recited over individual items during the meal.

The Hebrew word means 'bring out' or 'draw out'. It is a shorthand way of referring to the blessing recited over bread at the beginning of a meal, in which the King of the Universe is praised as the one who "draws out" bread from the earth, with the words "... haMOTzi LEchem MIN haAretz".

One of the 3 pieces of Matza - usually the middle one is used for Yachatz - the 4th part of the Seder - where the middle Matza is broken into 2 uneven pieces. The larger piece is kept for the Afikomen-dessert and the smaller piece - symbolizing Matza as the bread of affliction is kept on the Seder plate. Every meal on a Shabbat and Jewish festival requires 2 whole loaves of bread to be used during HaMotzi - the blessing over bread. Since on Passover bread is substituted with Matza, so we need 2 whole Matzot on the Seder plate. So we need 3 Matzot on the Seder plate: 2 for the HaMotzi, and one to be broken.

If you eat a meal after havdalah which includes bread, you say hamotsi. Kiddush is not said on Saturday night unless a Yom Tov begins on that night.
The traditional meal eaten after Havdalah includes bread (where possible), and is called the Melaveh Malkah, in honor of the departing Sabbath.