Firstly, yasher kochacho on a really good observation!
Rav Yosef Tzvi Rimon makes an interesting analysis here.
He notes that there are interesting parallels with the story of Sodom and Pesach which is highlighted by this constant repetition of "door".
On the pasuk in Bereishis 19:3 -
וַיִּפְצַר־בָּ֣ם מְאֹ֔ד וַיָּסֻ֣רוּ אֵלָ֔יו וַיָּבֹ֖אוּ אֶל־בֵּית֑וֹ וַיַּ֤עַשׂ לָהֶם֙ מִשְׁתֶּ֔ה וּמַצּ֥וֹת אָפָ֖ה וַיֹּאכֵֽלוּ׃ -
Rashi clearly states that he was baking these matzos as it was Pesach (referencing Bereishis Rabbah 48:12).
What follows is Lot is stuck in house with the malachim guarding him, then rescuing him and overturning Sodom. A close examination on the pasukim shows incredible similarities between what happened in Sodom and the Bnei Yisroel's salvation from Egypt.
Rav Rimon asserts:
The parallels attest to an amazing similarity between the two events: a closed house, a Destroyer, angels that destroy the wicked and save the good, departing from a certain place in the wake of a command (“Up, get out”), not delaying (Lot, unlike the Israelites, delayed and the angels had to drag him out themselves), and events that take place throughout the night.
So the first raft of "door" emphasis mirrors the experience in Mitzrayim where the Jews painted blood on the door frame and lintel so that the Malach Hamoveis would 'pass over' their door and target only the Egyptians. Equally, here the place of Sodom had been decreed to be destroyed through a heavenly force (i.e. angels) and Lot was safe on the other side of the door. Indeed (l'fi ani da'ati) this tallies with the Radak's definition of door on 19:9 -
הדלת. הִיא הַסּוֹבֶבֶת לִנְעֹל וְלִפְתֹּחַ - הדלת
THE DOOR — This term דלת signifies the wooden frame-work which swings round on hinges to close and open the entrance (cf. Verse 11).
A second parallel Rav Rimon notes is:
Both called for performing an act that endangered the one who did it, thus attesting that the person had rejected the ways of his environment: Lot took a risk by taking in guests and thus demonstrated his rejection of the society of Sodom; the Israelites slaughtered a lamb, an Egyptian deity, thus showing their severance from Egypt.
So, (again my analysis) the repetition of "door" helps bring this out. Sodom stood for being anti-hospitable, i.e. this meant not opening up your homes to guests. Yet, this mindset was antithetical to Lot who grew up in the house of Avraham and was regularly used to an 'open door' policy.
EDIT
As far as why there has to be this connection with Pesach, I would posit that possibly because the whole message of Matzah (i.e. poor man's bread) is on Pesach to remember that we are now free1 - we were saved and were zoche to a miraculous redemption. So perhaps in much the same way, we focus on it being Pesach by Lot because he too experienced a miraculous redemption - and the repetition of 'doors' helps to draw parallels.
Using this a baseline, perhaps it is this connection which ties in with the Haftarah. In the same way Lot was afforded Divine intervention similar to that of the Pesach story through agents of Hashem, we likewise have two instances where both a widow and the Isha Shunamis are sent a heavenly salvation through a shaliach i.e. Eliyahu HaNavi. The repetition of door helps to demonstrate as it was the channel through which these miracles were wrought. Indeed Rashi on Melachim II 4:4 writes that the reason why the door had to be closed was because it afforded respect to the ensuing miracle if it comes about in secret - i.e. it was a personal redemption that required a degree of modesty and privacy for the recipient.
1 Refer to Rav Yaakov Ettlinger, author of the Aruch LaNer in his Minchas Ani on the Haggada who writes; "The Seder opens with a declaration that we are about to eat “the bread of affliction.” We also invite those who are hungry and needy to join us in telling the story of the Exodus. There are two sides to matzah both literally and figuratively: it represents slavery and freedom, oppression and liberation."
