Q:

Sukkah: Why not tough it out? 

Why specifically by Sukkah is there a concept of מצטער פטור מן הסוכה הוא (Shulchan Aruch OC 640:4) someone who is in pain in by being in the Sukkah is exempt?  We don't have such a low threshold for most other mitzvos.    
Note: I understand the gemara Sukka (28b) makes a drasha - תשבו כעין תדורו מכאן אמרו כל שבעת הימים עושה אדם סוכתו קבע וביתו עראי - for the threshold, but why is this mitzva different? 

A:

As the Mishnah Berurah there explains, it’s because of תשבו כעין תדורו - one lives in the sukkah as he would in his house. 
You would leave your house if it was causing you significant discomfort, so you may leave the sukkah under the same circumstances. 

A:

I would like to potentially challenge the premise of the question.  You assume that the threshold of "someone who is in pain" only applies to the mitzvah of succah.  However, R. Joseph Messas has a responsum where he seems to apply this rule beyond the mitzvah of succah.  The question he was asked involved a certain case where two couples needed to get married but it was during the time period of sefirat ha'omer when weddings are customarily not held.  One of the arguments that R. Messas advanced to permit the weddings in this case was that not being married is considered "a great pain", and if one is exempt from a biblical commandment (succah) on account of pain, one is certainly exempt from a mere custom on account of pain:
Mayim Chayim 1:197

ועוד דעדיפא  שהרי אף במצות סוכה שהיא דאורייתא אמרו רז"ל מצטער פטור מן הסוכה כמ"ש בסוכה דף כ"ו ע"א וכ"פ רבינו הטור ומרן ז"ל בש"ע בסי' תר"ם ע"ש וא"כ כ"ש הכא שאינו אלא מנהג לזכר אבילות ישנה ויש צער גדול לאדם להיות שרוי בלא אשה ואין לו מי שישמשנו פשיטא דדיינו להעמיד הלכתא כדעת מהר"ם לונזאנו והרפ"ח ז"ל והחולקים עליהם הפריזו על המידה בזה אחר המח"ר מקדושת עצמותם

If the exemption for pain was a unique facet of the mitzvah of succah then it would not necessarily follow that it can exempt one from a custom. 
However, this responsum was severely critiqued by R. Nosson Nota Leiter.  R. Leiter wrote that heaven forfend to use such an argument.  He pointed out that by using such an argument one can eat chicken and milk that got mixed together because it is only a rabbinic prohibition and it causes the person pain to be unable to eat his meal.  If such pain can override the biblical commandment of succah, it should certainly override the rabbinic injunction against eating chicken and milk, according to R. Messas's reasoning:
Shu"t Tziyun L'nefesh Chaya # 29

מש"כ בסימן קצ"ז לענין נשואין בספירה ונתן טעם אחד ק"ו מסוכה דהיא מ"ע
  מה"ת ופטור מצטער מכש"כ כאן מנהג בעלמא דמותר משום צער ע"ז אני אומר ח"ו
  להשתמש בק"ו זה כי יצא מזה קל לחומר ואם יתערב בסעודתו בשר עוף וחלב נאמר
  שהוא מצטער לאסור לו סעודתו והוא רק איסור דרבנן נתיר לו לאוכלו ח"ו לא
  תהא כזאת בישראל

R. Leiter went on to state that the exemption of pain is a unique feature of the mitzvah of succah, and even by tefillin someone in pain is only exempt because the pain will prevent him from properly concentrating on the tefillin which is a requirement. 

וסוכה היא מצוה מיוחדת שבה נאמר הקבלה כי מצטער פטור ואם מצינו גם
  בתפילין הלא נאמר טעמו מפני שאסור להסיח דעת מהם ומצטער א"א שלא להסיח
  דעתו מהם

However, R. Leiter might not have been aware that this argument was not a novel idea from R. Messas.  In fact this very argument was used by no less than R. David Ibn Zimra in a similar responsum.  The question there dealt with cutting hair on Rosh Chodesh Iyar.  One of the arguments advanced to permit this was that refraining from cutting hair during sefirat ha'omer is merely a custom, and having overgrown hair is "pain", and if pain can exempt one from a biblical commandment (succah) it can certainly exempt one from this custom:
Shu"t Radvaz 2:687

תו איכא טעמא דאיכא צער בגידול השיער למי שרגיל להסתפר ולא עדיף האי
  מנהגא ממצות עשה של סוכה דקי"ל מצטער פטור

R. David Ibn Zimra actually went a step further in another responsum where he used the argument for an actual rabbinic obligation (as opposed to just a custom).  There the question concerned whether a sick person has to do shenayim mikra.  He answered that such a person is completely exempt from this obligation as since he is in the category of "one in pain" and he would even be exempt from the biblical commandment of succah, he is surely exempt from a rabbinic obligation. 
Shu"t Radvaz 3:425 (867)

ולענין החולה רואה אני שהוא פטור לגמרי ולא יהא אלא מצטער שהוא פטור מן
  הסוכה שהיא מצות עשה של תורה

Once again, if the exemption of "pain" was due to some unique aspect of the mitzvah of succah, the above argument would not necessarily follow. 

A:

This question is asked by the Be'er Yosef to Leviticus 23:42-43.  Here's a summary of what he writes, with some extra sources. 
We find two seemingly contradictory aspects to the mitzvah of sukkah.  On the one hand, it connotes exile.  Some explain 1 this is the rationale for why the mitzvah of sukkah is juxtaposed to Yom Kippur.  Just like Yom Kippur atones for a person’s sins, so does exile.  A person should atone for their sins by exiling themself to their sukkah.  In fact, if G-d forbid a person was decreed with exile due to their sins, they can “fulfill” this decree by exiling themself to their sukkah.  We even see this in the laws of sukkah itself.  If a person lives all year in a sukkah, they can’t fulfill their mitzvah by living there during the festival of Sukkos 2.  They must leave this structure and enter a different sukkah.  We see there is an aspect of exile associated with the mitzvah of sukkah.  This idea strengthens the above question.  Why is a person exempt from the mitzvah if they are in pain?  If anything, staying there would be a greater manifestation of the aspect of exile associated with the mitzvah of sukkah 3. 
On the other hand, the mitzvah of sukkah expresses joy and pleasure.  The Midrash 4 points out that the Torah mentions no idea of joy with the festival of Pesach, only once mentions it 5 with the festival of Shavuos, yet mentions it three times 6 with the festival of Sukkos 7.  This is because we are overjoyed after Yom Kippur that our sins have been atoned.  As well, Sukkos coincides with the harvest season, where a person is overjoyed at the bounty they have received from Hashem.  Many authorities 8 even hold that the physical pleasure one receives from dwelling in the sukkah is the essence of the mitzvah and not just a tangential aspect.  This seemingly contradicts the idea that a sukkah connotes exile, which is associated with pain and discomfort. 
However, these two ideas are not contradictory at all.  In fact, they are one in the same, with the same intent and manifestation.  The Torah wants us to be exiled from our permanent dwelling places and to dwell in the sukkah.  Yet we are obligated to experience great joy living in the sukkah, enjoying the physical pleasure it provides.  All of this is to remind us of the great chesed that Hashem did for us when He took us out of Egypt.  In essence, the Jews were exiled from their then homeland of Egypt.  Usually when a nation is exiled from one land to another, their experiences are full of pain and suffering.  They are starving, dehydrated, their clothing becomes ragged, they can’t sleep, etc.  This wasn’t so when they Jews were taken out of Egypt.  Hashem surrounded them with the Clouds of Glory, which protected them from the elements of the wilderness.  They were given the mun, the manna from Heaven to eat.  They were given water from the well of Miriam.  The Clouds of Glory even cleaned their clothing that they were wearing 9.  They lacked nothing, as if they were living in paradise and not in the wilderness.  To remember all of these miracles, we were commanded to dwell in sukkos.  There, we are able to experience the seemingly contradictory ideas of exile and joy. 
This also explains why a person who is in pain is exempt from the mitzvah of sukkah.  If a person is in pain and stays in the sukkah, they are going against the intention behind the mitzvah.  They are also considered an simpleton, since they in fact missed the point behind the festival of Sukkos.  They should have been filled with joy, taking great pleasure in the sukkah.  Through this they would have remembered all the miracles that Hashem performed for the Jews when they left Egypt.  They would have then fulfilled the verse “in order for your generations to know that I placed the Children of Israel in sukkos when I took them out of the land of Egypt” 10. 
So we see that because of the unique nature behind the mitzvah of sukkah, that's why it has this special dispensation when there is pain. 

1 Maharil Hilchos Sukkos § 2, 4.  The source for this is Pesikta D’Rav Kahana pg.  457 (Mendelbaum ed.,  found in Appendix “Parsha Acheres” s.v.  באספך מגרנך ומיקבך), brought in Yalkut Shimoni Emor § 653.  See also Zohar Emor pg.  103b
2 See Tosafos to Sukkah 2a s.v.  כי עביד ליה, Sukkah 8a with Rashi, Rashi to Sukkah 14a, and Shulchan Aruch Orach Chaim 636:2
3 See Pri Tzaddik Sukkos § 18 for a different explanation
4 Yalkut Shimoni Emor § 654.  The source seems to be Da’as Zekeinim to Deuteronomy 16:15
5 Deuteronomy 16:11
6 Leviticus 23:40, Deuteronomy 16:14,15
7 Mishneh Torah Hilchos Lulav 8:12 and Zohar loc.  cit. ( see the commentary of Mikdash Melech ad.  loc.)  also express that Sukkos has more joy associated with it than any other festival
8 The gemarra in Nedarim 16b says if a person forbids from themself the pleasure of dwelling in a sukkah (by making a neder), they are still allowed to dwell in their sukkah during Sukkos.  The gemarra says this is because of the rule that mitzvos were not given for physical pleasure, so the neder isn’t valid.  The Machaneh Ephraim Hilchos Nedarim § 25 asks based on Tosafos in Rosh Hashanah 28a s.v.  המודר הנאה (who say even without the mitzvah of sukkah there’s physical pleasure dwelling there), the pleasure would seem to be tangential to the mitzvah.  Therefore, the rule that mitzvos were not given for physical pleasure shouldn’t apply.  All that says is the fulfillment of a mitzvah isn’t considered physical pleasure or benefit (Ran to Nedarim 15b).  Why then is someone who makes this neder permitted to dwell in the sukkah?  The Oneg Yom Tov Orach Chaim § 50, Teshuvos Chemdas Shlomo Orach Chaim § 23, and Teshuvos Rabbi Akiva Eiger II § 138 explain that the neder would work only when the pleasure is distinct from the fulfillment of the mitzvah.  Not true for the mitzvah of sukkah, where the pleasure of dwelling there is the mitzvah itself.  Therefore, it would be permitted to dwell in the sukkah despite the pleasure they receive from it, since the rule mitzvos weren’t given for pleasure would apply.  Even though the mitzvah of sukkah would seem to have been given for personal pleasure, perhaps there’s a difference between mitzvos being given for pleasure and mitzvos obligating pleasure
9 Rashi to Deuteronomy 8:7
10 Leviticus 23:43

