1) Just to make it a chazakah of 3 weeks in a row with R' Shteinman's torah: R' Shteinman has a safeik whether the kohen who rules whether a nega is tamei or tahor recites a bracha, something like, "v'tzivanu liros v'lifsok ha'nega." Beis din does not recite a bracha when they pasken, so why should this case be any different?
The difference is that the role of beis din is birur -- they are just there to clarify the facts, and m'meila what the din should be is clarified. The kohen's psak is not birur. The facts on the ground -- whether the afflication in question is a nega -- might be perfectly clear before the kohen says anything. However, until the kohen declares something to be a nega, it's like it does not exist. The kohen's psak does not merely clarify; it creates a new reality.
My 2 cents: whether b"d's function is just a matter of birur may not be so simple. The Rashba (B"K 90) writes that lo te'hey shmiya gedolah m'reiya only works for a beis din of three, but not a yachid mumcheh. From the perspective of ascertaining the facts, there is no difference between the re'iya of the mumcheh and the re'iya of a b"d of three. It would therefore seem that there is some extra ingredient involved in the mix. See Sharei Yosher 7:1 For a different approach, see also R' Amiel in his Darkei Kinyanim.
2) There is a derasha (Nida 41) on the second pasuk of our parsha, אִשָּׁה֙ כִּ֣י תַזְרִ֔יעַ וְיָלְדָ֖ה זָכָ֑ר:
א״ר יצחק א״ר אמי, אשה מזרעת תחלה יולדת זכר, איש מזריע תחלה יולדת נקבה, שנאמר אשה כי תזריע וילדה זכר.
The gemara at the end of Brachos (60) raises the possibility of a person davening for their child to be a boy, at least within 40 days from conception, and asks how this is possible given that everything is determined already at the moment of conception, by whether אשה מזרעת תחלה or איש מזריע תחלה:
והא"ר יצחק בריה דרב אמי איש מזריע תחלה יולדת נקבה אשה מזרעת תחלה יולדת זכר שנאמר אשה כי תזריע וילדה זכר הכא במאי עסקינן כגון שהזריעו שניהם בבת אחת
The dochak seems evident. The Zohar raises the same issue in a slightly different way, but the answer is unclear (to me at least):
אִשָּׁה כִּי תַזְרִיעַ. תָּנֵינָן, אִשָּׁה מַזְרַעַת תְּחִלָּה יוֹלֶדֶת זָכָר. רִבִּי אֲחָא אָמַר, הָא תָּנֵינָן, דְּקוּדְשָׁא בְּרִיךְ הוּא גָּזַר עַל הַהִיא טִפָּה, אִי אִיהוּ דְּכַר אִי אִיהִי נוּקְבָּא, וְאַתְּ אַמְרַת אִשָׁה מַזְרַעַת תְּחִלָּה יוֹלֶדֶת זָכָר. אָמַר רִבִּי יוֹסֵי, וַדַּאי קוּדְשָׁא בְּרִיךְ הוּא אַבְחִין בֵּין טִפָּה דִּדְכוּרָא וּבֵין טִפָּה דְּנוּקְבָּא, וּבְגִין דְּאַבְחִין לֵיהּ, גָּזַר עָלֵיהּ, אִי לִהֱוִי דְּכַר אוֹ נוּקְבָּא.
The difference between the Zohar and the gemara is that the gemara seems like a "G-d of the gaps" type argument. Most things are determined by the teva of whether אשה מזרעת תחלה or איש מזריע תחל, but where that's not possible, G-d steps in to take over. The Zohar sees G-d as taking a hand in everything and somehow ironing out there being a need for that alongside how teva works.
In short, we are left with this tension between whether the determining factor of things is G-d or teva, which the Chasam Sofer encapsulates nicely with the following question: We thank Hashem every morning שׁלא עשׂני אשׂה. What are we thanking Hashem for? אשה מזרעת תחלה יולדת זכר, איש מזריע תחלה יולדת נקבה, it's determined by nature?
Chasam Sofer answers that there is in effect a two track system. There is the derech ha'teva system of אשה מזרעת תחלה יולדת זכר, איש מזריע תחלה יולדת נקבה. On top of that, there is the system that Klal Yisrael operates under, the system that transcends teva, that allows for Hashem to step in and direct things.
He then reads this into the pesukim. The parsha starts אִשָּׁה כִּי תַזְרִיעַ וְיָלְדָה זָכָר, which is the normal progression of teva. The continuation וְאִם נְקֵבָה תֵלֵד וְטָמְאָה שְׁבֻעַיִם כְּנִדָּתָהּ is not the flipside case of איש מזריע תחלה and having a girl. Were that the case, ikkar chaseir, why not say אישׁ כּי תזריע...? Rather, the continuation is this second track, the l'maalah min ha'teva track. Even though al pi teva אשה מזרעת תחלה יולדת זכר, sometimes Hashem will redirect things. Even though it's a case of אשה מזרעת תחלה, it can still come out that וְאִם נְקֵבָה תֵלֵד... In such a case you might have thought that the dinei tumah should be like a זָכָר, like the case of אשה מזרעת תחלה, which is what happened, kah mashma lan that it doesn't work that way.
The lesson here does not just apply to our parsha. Al pi derech ha'teva all kinds of things are determined -- how much $ you will make, how good you will do in school, etc. -- but there is an override switch to everything. Only Hashem has the final say.
Maybe that's why this parsha of tumas leida appears as an introduction to the rest of our parsha, which deals with the issue of negaim. A nega can come as a result of disease; it can be the result of teva. But that does not make a person tamei. Without the declaration of the kohen, the nega is meaningless. There is the override switch that determines the true reality of what is and what will be.