Why is hallel at the seder different from hallel during the year?

Question

I know there are many differences between hallel at the seder and regular hallel.

At the seder, hallel:

1. Has no starting bracha (Orech Chaim Hilchot Pesach Siman 473)
2. Is said sitting down (Sefer Shibulei Haleket Inyan Rosh Chodesh Siman 373)
3. Is said at night (Hallel is supposed to be said during the day normally) (Bavli Megillah 20:2)
4. Hallel at seder is not part of the list of 18 times when a full hallel is said. (Bavli, Archin 10:1)

What is the reasons for these disparities/on what basis were they made in the haggada?

Answer 1

Answer to Q1 from here:

A. Some contend that, despite inferences to the contrary, Hallel on Seder night is not a mitzvah but only expresses our rejoicing (Shu’t Ri MiGash #44).

B. Alternatively, although there is a mitzvah Seder night to praise Hashem, this praise could be spontaneous and unstructured which would not technically require reciting the structured Hallel. Since no specific song or praise is required, Chazal did not require a bracha before singing Hallel (see Rav Hai Gaon’s opinion, as quoted by Ran, Pesachim Chapter 10).

C. Although Hallel Seder night should require a bracha, we cannot do so because we interrupt the recital of the Hallel with the meal (Tur Orach Chayim 473).

Answer to Q2:

Most mitzvos are performed while standing, and there are additional reasons why Hallel should be recited standing. Hallel testifies to Hashem’s miracles and wondrous deeds, and testimony must be made while standing (Mishnah Berurah 422:28). Furthermore, the pasuk in Hallel declares, “Sing praise, servants of Hashem who are standing,” implying that this is the proper way to give praise (Shibbolei Leket).

On the other hand, at the Seder Hallel is recited sitting, because this demonstrates that we are freemen (Shibbolei Leket).

Answer to Q3:

Chazal derive from the verse of Hallel, “From when the sun rises in the east until it sets shall Hashem’s Name be praised,” that Hallel should be recited by day and not by night (Megillah 20b). Although the day begins when the eastern horizon lights up (amud hashachar), Chazal ruled that Hallel should not be said until after sunrise.

The exception to this rule is when we recite Hallel on Pesach night as part of the Haggadah, since the miracle took place at night. Many communities have the custom of reciting Hallel in shul, also, that night.

Answer 2

Answer to Q4:

In the version in Maseches Sofrim (20:9), this teaching is actually cited in the way you propose:

דתני ר׳ שמעון בן יהוצדק ימים שמנה עשרה ולילה אחד יחיד גומר בהן את ההלל [...] ובגולה עשרים ואחד יום ושתי לילות

For R' Shimon ben Yehotzadak taught: On eighteen days and one night does one complete Hallel. [...] In the Diaspora, twenty-one days and two nights.

Likewise Tosefta Sukkah 3:2 (h/t magicker72):

י"ח יום בשנה ולילה אחת קורין בהן את ההלל

On eighteen days and one night does one complete Hallel.

And likewise Yerushalmi Sukkah 4:5 (19b) (h/t again magicker72):

תני שמונה עשר יום ולילה אחד קורין בהן את ההלל בכל שנה.

It was taught: On eighteen days and one night does one read Hallel on them every year.

As to why Arachin doesn't say it this way, presumably it's because the Gemara isn't discussing nights. In context, it's addressing days on which certain instruments were played in the Beis HaMikdash before the Mizbeiach; as this is only done during the day, there's no need to address the one or two nights on which Hallel is also said.