As stated, the custom is to refrain from reclining while reciting the Hagada, because it must be recited with seriousness and reverence (MB 473:71, based on Shlah). However, this seriousness incorporates joy and elation at the fact that God chose us from among all the nations and gave us the Torah (see Kaf Ha-ĥayim 473:152).
We have already seen that the matza must be kept uncovered while we recite the Hagada, in keeping with the words of the Sages: “’leĥem oni’ – bread over which we ‘onim’ – answer or say – many things” (Pesaĥim 115b). Giving concrete expression to the Exodus story is the foundation of the mitzva to eat matza on the Seder night.
However, when we raise the wine glasses to recite the paragraphs of “Ve-hi she–amdah” (“And this [promise] has stood”) and “Lefikhakh anaĥnu ĥayavim” (“Therefore it is our duty”), as well as when reciting the long berakha on redemption (“Birkat Ha-ge’ula”) just before drinking the second cup, the matza should be covered. The matza is more significant than the wine, and therefore, whenever we hold up the cup of wine and show it preference, the matza must be covered (SA 473:7, MB ad loc. 73). It is for the same reason that we cover the bread when reciting kiddush every Shabbat and Yom Tov.
When reciting the paragraph “Matza zo she–anu okhlim” (“This matza that we eat”), the Seder leader holds up the matza for all of the participants to see, in order to endear the mitzva to them. And when “Maror zeh…” (“This maror…”) is said, the maror is held up. However, when “Pesaĥ zeh…” is said, the zero’a is not held up, because it is not the actual meat of the korban Pesaĥ, but merely a commemoration of it. Thus, one who holds it up is like one who offers sacrifices outside the Temple precincts (Pesaĥim 116b; SA 473:7).
It is customary to spill out a bit of wine from the cup when enumerating “dam, va–esh, ve-timrot ashan” (“blood, fire, and pillars of smoke” – Yoel 3:3), while reciting “detzaĥ, adash, and be’aĥav” (R. Yehuda’s mnemonic device for remembering the Ten Plagues), and while enumerating the Ten Plagues. This comes to sixteen times. Some have a custom to drip the wine with the index finger, and others have a custom to pour out a little bit into a broken vessel (Rema 473:7; SHT 81 and Kaf Ha-ĥayim 163-4 ad loc.). Rav Kook rules that one should not spill out Shemitta wine (produced from grapes grown on the Torah’s Sabbatical year; some poskim say that there is a mitzva to drink Shemitta wine, and therefore one should not refrain from using it for the four cups).