Rambam writes in his Introduction to The Mishnah:
Afterwards, he [Rabbi Yehudah HaNasi] divided the material in the Order of Women [Nashim]. And he started with Tractate Yevamot (Levirate Marriages); and the reason that required him to start with Yevamot and not to start with Tractate Ketubot (Marriage Contracts) – and the investigation of the intellect ...
The first Tosofos Yeshanim in Mesechtas Yevamos asks why Yevamos comes first. We don't start with tragedies so Kiddushin should be first.
He gives two reasons for why Yevamos was put first
(1) Because Seder Nashim follows Seder Moed. Seder Moed ends with Moed Kotton (according to their ordering) so we put and study the tragedies near each other.
(2) Because ...
The OU website notes a good reason:
While Masechet Yevamot focuses on the rules and regulations concerning levirate marriage, nevertheless, it involves an examination of forbidden sexual and marital relationships, as well as the severity of those prohibitions. Thus, the study of Masechet Yevamot becomes a basic source for all of Seder Nashim.
The Talmud states (Sanhedrin 17a) that one can only be appointed to the Sanhedrin if he can prove that a sheretz is pure (despite the fact that a sheretz is impure):
אמר רב יהודה אמר רב אין מושיבין בסנהדרין אלא מי שיודע לטהר את השרץ מה"ת
Rab Judah said in Rab's name: None is to be given a seat on the Sanhedrin unless he is able to prove the cleanness ...
Rambam, the most prolific rishonic author of laws of Zeraim, writes as follows in his introduction to Mishneh Torah:
ומשני הגמרות ומן התוספתות ומספרא וספרי (ומן התוספות) מכולם יתבאר האסור והמותר הטמא והטהור החיוב והפטור הפסול והכשר כמו שהעתיקו איש מפי איש מפי משה רבינו מסיני
From the entire [body of knowledge stemming from] the two Talmuds, the Tosefta, the ...
You understand that “for the sake of heaven” implies to “have influence on Heaven's sake one way or the other, for good or for bad ?” You see an action undertaken for the sake of heaven meaning that the action has influence on heaven.
Consider מחלוקת לשם שמים
Wikipedia defines it as
מחלוקת לשם שמים היא מחלוקת עניינית, שהעוסקים בה חותרים להגיע לחקר
One way to look at it, is the notion of going above and beyond, transcending the normal human existence to put Hashem and his Torah/mitzvos first.
As you mention, there are numerous examples, but perhaps we can start with one in Pirkei Avos 2:12 -
רַבִּי יוֹסֵי אוֹמֵר, יְהִי מָמוֹן חֲבֵרְךָ חָבִיב עָלֶיךָ כְּשֶׁלָּךְ, וְהַתְקֵן עַצְמְךָ לִלְמֹד תּוֹרָה, ...
For no reason he did give an improbable explanation, it's to say, he was thinking that there is a kashia and he forced himself to give an explanation far from the simple sense because the speaker has an okimta regarding the timing of the situation.
Anyway, there is always a choice between two dochakim. But the speaker prefers to maintain a simple pshat in ...
When a man marries a woman he says הרי את מקודשת לי כדת משה וישראל, "You are married to me according to the law of Moses and Israel." In other words, he only wants the marriage to go into effect if it is consistent with the laws of the Torah (דת משה) and other rabbinic decrees (דת ישראל).