Bava kama 9b - hiddur is an increase of 1/3 of what?

Tosfos suggests that the 1/3 increase is not in cost but in the size of your esrog that if you have one that is the size of a nut, then increase by 1/3. The Beis Yosef explains this idea based on others, that the meaning of Tosfos is that you are only being asked to do this hiddur of a 1/3 if you are in a state where you have the bare minimum of the mitzva. Then you should try to increase 1/3. But if you have a decent size esrog, then no 1/3 increase is necessary.

bava kama 9b - Hiddur Mitzva

There is a famous chakira about the din of hiddur mitzva, זה א-לי ואנוהו if it is a separate halacha we apply to mitzvos or if it is incorporated into the mitzva itself. Rav Soloveitchik explained that this chakira explains the two side of the question in our gemara of the 1/3 hiddur is מלבר or מלגיו. He explained that if we say that it is part of the mitzva, then it makes sense that when we look at the chefza you have at the end, that 1/3 of it be hiddur, thus it would be מלבר but if it is just a separate halacha then using what you have now, you tack on the hiddur of 1/3

Bava Kama 6a - nezikin learned out from 2 avos

There is a machlokes rishonim what is the status of Nezikin that are learned out from multiple avos.
This is discussed in the Rosh and expounded in the Griz al Harambam in the beginning of Hilchos Nizkei Mammon.
The question is as follows: אבן,סכין ומשא  that fell down and cause damage after landing, we learn according to Shmuel that if the owner was not mafkir, then it is learned out from a combination of esh and bor.
Tosfos understands this to mean that it will have the halachos of what it was learned from, i.e. this will have on the one hand a petur of tamun like esh and on the other hand, like bor that adam and and keilim are patur.
However, there is an alternative understanding to this The Rosh explains that אבן,סכין ומשא, once the learning is complete from esh and bor, are then completely like bor. Esh was used to learn out that one is chayav but it is a pure bor in terms of halacha. This actually also works well with what we learned in shiur that even according to Rav, he also really learns out even sakin umasa from a combination of bor and shor (mentioned as shitas Rashbam in tosfos 3b), just acc to him, after we learn it, they are pure shor as opposed to bor in terms of the halacha. And according to Shmuel, it is a pure bor after the derivation.
The Griz explains this approach as follows: the petur of tamun is a גזירת הכתוב and limited to only an item that has a שם אש but something that אש was used to learn it out does not make it אש and therefore the petur does not apply. This would be true in a reverse case as well, where the item is defined as esh and bor was used to learn it out, i.e אבן,סכין ומשא that cause damage while airborn. However, Tosfos disagrees and says whatever is the source to learn out the din that you are chayav, must also reflect on the parameters of what you are chayav for.
The Rosh's explanation also enables us to have an alternate peshat to Tosfos 3b on top that says we can only learn from keren to even sakin umasa since only keren is chayav in רשות הרבים. But if we say that the נלמד does not have to have all the halachos of what it learned from applied to it, then we are ok to learn even from shen or regel.
Similarly, the Griz explains that תולדותיהם כיוצא בהם או לאו כיצא בהם according to the Rosh is really asking the following: do the toldos of the avos have the exact status as the avos in terms of all the halachos that are unique to each one or not, or are they merely learned out from the av but they dont have the same שם as the av and therefore would not have all the unique dinim. And the answer of the gemara is that since all the toldos have all the characterstics of the av, therefore they have all the unique halachos as well.

Bava Kama 3a -- categorizing a snake bite

The gemara states that being bitten by an animal is a תולדה of שן. However, Tosfos points out that this would not include a snake bite which would be a tolada of keren. This relates to the sugya in Arachin 15a we learned where the animals discuss with the snake how it does not get הנאה from its' bites.
However, the Rashba says that a snake bite can still be a tolada of שן. The reason for this is that the snake still has הנאה even though it does not digest. Thus this explanation still works with the gemara in Arachin since there it discussed eating and not biting.
Just to round out all the possibilites, Tosfos on 16a says that a snake bite is a תולדה of רגל since this is a normal act of a snake and Tosfos therefore says that if it is רגל this would mean that you would be פטור in reshus harabim if your snake bites someone!

kerisos 8a - takanas Rabban Shimon Ben Gamliel

Rashba"g made a takkana in order to drive down the price of the birds for korabanos, that a woman is not chayav to bring korbanos for all vaday births. This is very difficult to understand -- how can he go against a chiyuv in the Torah for korbanos? See the Shita Mekubetzes in the back who bring the peshat of Rabbeinu Tam.

The Netziv in the Harcheiv Davar at the beginning of Parshas Tazria also grapples with this issue of Rashba"g seemingly going against halacha. To solve this problem, he says in the name of his son, Rav Chaim Berlin that we can see the peshat of Rashba"g in the pesukim of Tazria. The summary pasuk of "Zos Toras Hayoledes" is before the pasuk that addressed the poor korban. One would have thought that the order should have been reversed. Based on this, he argued that the ruling of multiple korbanos for multiple leidos is only for a rich yoledes. However if she is poor, not only does she bring a bird instead but also she only has to bring one for multiple births. Therefore when it became expensive, Rashba"g said the rest of the korbanos are no longer obligated and the yoledes only has to bring one, but should she become rich, then she would be obligated in all of them.

Thanks to my Rebbe, Rav Binyamin Tabory, may HKBH send him a refua sheleima, for pointing this Netziv out to me.

Shabbas 130a - Was Elisha Baal Kenafayim a model of Mesirus Nefesh?

According to the simple peshat of our sugya, the gemara is saying that we are not moser nefesh for tefillin the way that Elisha Baal Kenafayim did. But Tosfos on 130a quotes the Rashbam who says the opposite. When Elisha answered the officer and did not say he had tefillin, this was a proof to the claim of the gemara that no one is moser nefesh for tefilln, since even he did not and instead said he had kanfei yona.

The Maharam explains that the reason The Rashbam did not follow Rashi’s peshat is that Rashi implies that you need everyone to be moser nefesh and here only one person did. However, this is not true since even by Avoda Zara we find most did not other than Chananya Mishael and Azarya.

So why did a miracle happen to him? Perhaps because he was dedicated to the mitzva in a way that others were not in terms of being makpid on Guf Naki?

Shabbas 117b – Bahul al mamono

Rava explains the limit of how much we are allowed to remove from the house that is burning to be based on מתוך שאדם בהול על ממונו אי שרית ליה  אתי לכבויי.

This is contrast to 43b where R’ Yehuda Ben Lakish says מתוך שאדם בהול על מתו אי לא שרית ליה אתי לכבויי. Tosfos there (44a) explains that a person has more concern over a dead body than his money so he will put it out on purpose if we dont allow him a way to save the body. The concern over money however is more of when he is allowed to save, he may get so involved that he will forget its shabbas and then put out the fire.

However, the sugya on 65b still needs to be resolved with our sugya. The gemara, when raising the issue of saving clothes from a fire (הערמה), says that according to Rabbi Meir  on 120a we need to allow all clothes for if not,  דאי לא שרית ליה אתי לכבויי, which is like the death case and not monetary case above!

The Aruch Hashulchan in Siman 334 explains the difference between the sugyos as follows. He says that Rashi in 117b writes “ואי שרית ליה לאתחולי בהצלה”. This word is the key to understand, as it is based on human psychology. When a person is heavily involved in the act of saving things due to the fire he may forget its shabbas and chazal need to consider that and therefore use behala to expand the heter and allow him as much as possible so he does not resort to putting out the fire. Once he has been given enough leeway to act, he is calm and focused and we are no longer concerned about him putting it out, but if we limit him partially, then he may have more confusion and forget its shabbas. On the other hand, sometimes we may limit him from getting involved in kibuy, as in saving three meals which is a small act and not even a “start” in Rashi’s words, then the person understands that Chazal limited what he can do and he accepts it as a gezeira mishamayim and the worry is not that he will put it out, and instead we need to keep him away from getting involved in the acts that would make him forget its shabbas.

Therefore, according to Rabbi Meir we must allow all clothes and not limit to 18 as we are allowing him to act, and similarly on 120b, Chachamim allow all vessels including kelei cheres that will explode immediately since once he is allowed to build the mechitza we need to allow him to finish, for if not, this may cause behala and putting out of the fire.

The Aruch Hashulchan does not address Rabbi Yossi on 120b who says the opposite sevara – that by mechitza, he says that “אי שרית ליה אתי לכבויי”- which would seem to indicate that Rabbi Yossi and Chachamim are arguing about the Aruch Hashulchan’s explanation. Perhaps Rabbi Yossi has a different point of argument and is arguing on how sharp the lines have to be drawn between gerama and kibuy and his concern is that we cannot allow gerama because it looks too much like kibuy and this is unrelated to the question above.