Why do we say baruch hamokom baruch hu?
Why that name of hashem here? There are many more names we could have used.
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The ריטב"א, in his commentary on the haggadah, says in his second answer (after explaining that this is the beginning of the part where we darshen psukim, and that we need a quasi-birchas hatorah first, which is baruch hamakom, and why did we use a lashon of makom which implier rulership and existing over more than the whole world)
וי"א כי אמר מקום בענין זה לפי שעד שלא נתן הקב"ה את התורה לישראל לא היה נודה הוא ולא חכמתו, אלא כדבר הגנוז במקום נסתר ואינו נראה וכשנתן תורה לישראל אז הופיע אורו לכל העולם
And you could say that it says the title מקום here (by this birchas hatorah) because before the Rebbono shel Olam gave the Torah to the Jews neither he nor his wisdom were known, rather it was hidden, but once he gave the Torah his light spread to the whole world (and thus, HaMakom - one who is in every place, and known there)
(My own very loose translation)
In short, at risk of being repetitive - this is a birkas hatorah, and the Torah is what spreads the Rebbono shel Olam's light on the whole world, which is what the title HaMakom represents.
Regarding the type of question "why this and not that?":
Antoninus said to Rabbi, ‘Why does the sun rise in the east and set in the west?’ He replied, ‘Were it reversed, thou wouldst ask the same question.’ (Soncino translation)
Guide for the Perplexed 3:26 (Rambam)
You ask why must a lamb be sacrificed and not a ram? but the same question would be asked, why a ram had been commanded instead of a lamb, so long as one particular kind is required. The same is to be said as to the question why were seven lambs sacrificed and not eight; the same question might have been asked if there were eight, ten, or twenty lambs, so long as some definite number of lambs were sacrificed. It is almost similar to the nature of a thing which can receive different forms, but actually receives one of them. We must not ask why it has this form and not another which is likewise possible, because we should have to ask the same question if instead of its actual form the thing had any of the other possible forms. Note this, and understand it. (Friedlander translation)
Wars of the Lord Introduction (Ralbag)
Hence the reader should not inquire concerning these things why we have treated this thing before some other thing, since [he thinks] that the other thing should be treated first because of one of the aforementioned reasons. We have in fact treated one particular subject first precisely because of one of these reasons: and it is obvious that if we had adopted the reverse order the same question would have been raised. (Feldman translation)
Regarding specifically the usage of different names of God:
Emunos V'deios 2:3 (Sa'adia Gaon)
ואם יאמר אדם מה ענין שני שמות הללו המשומשים תמיד במקרא ה' אלהים נאמר כבר קבע שהם לענין אחד כאמרו כי כה אמר ה' בורא השמים הוא האלהים ואמרו דעו כי ה' הוא האלהים הוא עשנו וגו' ואחר הקביעה הזו אין לחוש אם יתאר אחד מהם בפעולה אחרת ודומה לזה בלשון וילך ירובעל בן יואש וישב בביתו ואמר אחריו ולגדעון היו שבעים בנים יוצאי ירכו תיאר שם זה בפעולה והשני בפעולה אחרת ולא חש כיון שכבר קבע בדבריו כי ירובעל הוא גדעון (Kafih translation p. 87)
Now if someone were to ask: "But in that case what is the meaning of these two names 'Ădhonay and 'Ĕlohim that are constantly employed in the Bible with reference to God?" our answer would be that Scripture makes it quite clear that they both have one connotation. This is borne out by its statement: For thus saith the Lord that created the heavens, He is God (Isa. 45:18), as well as its statement: Know ye that the Lord He is God (Ps. 100:3).
After this definition, then, no attention is to be paid to the fact that one of these two appellatives is used in the description of one action whilst the other is used in describing another, for there is many a parallel to this usage in the language of Scripture. Thus it states: And Jerubbaal the son of Joash went and dwelt in his own house (Judg. 8:29), and after that: And Gideon had three-score and ten sons of his body begotten (Judg. 8:30). One name is here used in the description of one act and another in connection with another without any scruple, because it has already been definitely stated that Jerubbaal was identical with Gideon. (Rosenblatt translation p. 99)
The implication is that once we know God's names it makes no difference which name is used in which context. Indeed, Rosenblatt in his footnote to the above passage writes:
Saadia undoubtedly has in mind such statements as the one quoted in Genesis Rabba (xxxiii) on genesis 8:1 to the effect that the name 'Ădhonay stands for God's quality of mercy, whereas that of 'Ĕlohim betokens His quality of justice.
Hashem is called hamakom because He is the place of the world rather than the world being His place.
In other words, one of the problems Jews have faced over the past 3000 years is there is no place for them and they are squeezed out of existence.
So we thank Hashem for redeeming us and creating 'space' that we exist in as His people.
Rav Schwab in the back of his sefer on tefilla (p.549) explains HaMakom is the same lashon used for aveilus, as in hamakom yinachem. The reason for this is because at this point the Haggada references the four sons, not necessarily where you hoped they would be holding, but you accept the lot HaMakom gave you.
This seems to be a similar idea to the insight msh210 shared regarding HaMakom by a mourner.
The name Hamakom refers to the fact that Hashem includes the whole world. We invoke this name by comforting a mourner, perhaps because comfort comes with the big picture. In the Haggada we are noticing how all-encompassing the Torah is, in that it relates to all four types of children. Therefore, we address the all-encompassing G-d who gave it.
See Emes LiYaakov on parshas Vayeitzei on the words Vayifga Bamakom. He brings various Talmudic proofs that during conversation, referring to Hashem as 'Makom' works as a stand in for an actual name of Hashem, and one can fulfil saying a bracha with it. This term would not work for an actual bracha said in the context of making a blessing.
In footnote 5 there the writer mentions that Reb Yaakov used this to explain the phraseology Baruch Hamakom Baruch Hu in the hagada. Namely, the entire hagada is said derech sippur, as a conversation. Therefore we are not allowed to mention the name of Hashem, as that can only be done while learning Torah. Therefore, when the author of the hagada wants to bless Hashem he refers to Him as 'Hamakom' or as 'Shenasan Torah liAmo Yisroel', but doesn't mention Hashem. So too in the entire hagada one should not mention the name of Hashem, (except in an actual passuk). And this was in fact the minhag of Reb Yaakov Kaminetzky.
In the Haggada of Rav Pam (p.56), he quotes Rav Shmuel Avigdor Feivelsohn (Rosh Kollel Nachalas Naftali) who had an encounter with a Russian immigrant on a bus in Bnei Brak.
The jew told him, “Lenin taught the Russians that the Jews aren’t a nation because they need a common language, culture, and land – yet Jews don’t have their own land. But Lenin is wrong because there IS a place for all of Klal Yisroel – wherever we find ourselves in life, Hashem is our makom. He is always our place, so we are a nation with a place of our own.” Hence, Shemos (19:5) וְעַתָּה אִם־שָׁמוֹעַ תִּשְׁמְעוּ בְּקֹלִי וּשְׁמַרְתֶּם אֶת־בְּרִיתִי וִהְיִיתֶם לִי סְגֻלָּה מִכָּל־הָעַמִּים כִּי־לִי כָּל־הָאָרֶץ.
Seemingly, HaMakom in the Haggada refers to Hashem as the common "land" which gathers together all of Klal Yisroel in order to be considered a nation with a language, culture, and land.