Is there any halachic reason why we do the seder in the order that we do? It seems that many of the components of the seder are halachically mandated, but is there any halachic mandate that we do them in a particular order, or is this just a custom that has arisen?
The order of the seder is discussed in Shulchah Aruch, see Simon tuf-ayin-beis Dinei Haseva and Daled Kosus (472, laws reclining and the four cups), tuf ayin-gimmel, dinei kos rishon v'seder hapesach until kos sheni (473, laws of the first cup and seder of pesach until the second up), and so forth until tuf-peh-aleph, sh'lo l'shtot achar arbeh kosos (481, not to drink after the four cups).
See Shulchan Harav (the same simonim), now available in English, for a concise presentation. Note also, that in some instances, doing an item out of order, can create a halachic issue.
Here is a short explanation of the order:
Kadesh is necessarily the first thing, in any yom tov or shabbos meal.(Shulchan Aruch 472:1 and Magen Avraham, Sulchan Aruch Harav, 472:1)
Urechatz - we have to wash before handling wet food (without making the beracho netilas yadayim, see the discussion in Shulchan Aruch 473:6, T"z 7, Sh. HaRav 473:4).
Karpas - we awaken the child's curiosity before we start the story,
Yachtza - we tell the story over a poor man's bread, we need that the others remain whole to make the berachos later
Maggid - we are obligated to tell the story, fulfilling the passage when your child asks, and while the Matzoh is in front of us (See Mishnah Pesachim 10:4 regarding the questions and outline).
Rachtzah - now we are going to eat the matzo, we have to wash with a beracha
Motzi - we have to make a beracho hamotzi to eat the matzo
Matza - we make a beracho on the mitzvoh we are about to perform, eating the matzo, it is not an interruption after hamotzi, because it is required for the eating.
Maror - another obligation, matzo and maror
Koreh - we are still working on the requirement matzo and maror
Shulchan Aruch - now we can eat, and we need to eat a festive meal because it is yom tov
Tzafun -this is in place of the lamb, which is to be eaten on a full stomach
Beirach - benching after the meal
Hallel, Nirtzah - we started this in the haggada. The first part refered to the redemption from Egypt. Here we say the part that refers to the miracles that will happen in the future (Pesachim 118a, Zevach Pesach) and again, we sing praises over a cup of wine, and finally, pray that our service be accepted.