bava kama 57b - petur for a shor פקח that fell into your bor

The Gemara's conclusion is that if a שור פקח falls into your bor during the day, then you are patur.  However, the Rambam rules in 12:16 in Nizkei Mammon that if the shor only got damaged then the owner of the Bor would be Chayav.

The Raavad asks the obvious question -- if the shor is פקח then what is the basis to differentiate between damage and death? You should be completely patur for this שור פקח. Interestingly, The Raavad is ok with the Rambam stating that for a person who gets HURT, that you would be chayav even though for the animal you should not.

Explaining the Raavad is simple, since we have learned about people אין דרכן של בני אדם להתבונן בדרכים so this leads to a chiyuv for a person that falls into the Bor while still being patur for a שור פקח . The Maggid Mishne explains that perhaps since a person thinks so he does not look around as carefully as a Shor.

So, now we just need to understand the Rambam.
The Maggid Mishne explains that this פקחות of the animal only means that it avoids a serious pit that could kill it but its not smart enough to avoid all damaging obstacles. Based on this, the question arises -- what if the animal falls into a bor that COULD have killed it but only got hurt. Using this logic, do we say that the בעל הבור is patur? The Aruch Hashulchan (choshe mishpat 410:24) says yes. Minchas Chinuch also thinks that this would be correct but says there is no indication of this in the Rambam so he writes ויש ליישב

Bava kama 44b (and 13b) - damage done by shor hefker

We see on 44b that according to חכמים we kill the animal that is hefker and Rabbi Yehuda disagrees.
When we learned 13b, we saw Rabbi Yehuda's opinion there but it was quoted by Ravina, so one could have thought that this was according to everyone (even though Rabbi Yehuda is brought there as proof for this).
The Rambam paskens both sugyos. In 8:4 he paskens that a shor hefker is patur when it does damages using pasuk of שור רעהו. But the Rambam paskens like חכמים in our sugya and (10:6) and says that we kill a shor hefker
The Lechem Mishne and others point out that there is no contradiction. The sugya on 13b is discussing damage while our sugya discussed death. Even though Rabbi Yehuda is brought as proof by Ravina, the petur of shor hefker by nezikin is according to everyone.
Perhaps we can suggest that according to חכמים , when we kill the shor we are not killing it as a punishment for the owner but to remove a dangerous being from the world but in the case of nezek, if there are no owners we have no one to punish. Rabbi Yehuda could be saying that even in killing, the focus is on the owner who did not guard.
However, this explanation does not cover the case of a shor that damages and then the owner is mafkir, where even here the former owners are not liable.

bava kama - mishna 19b -- is meshune keren?

The Mishna on 19b begins a discussion about שן. It then discusses eating כסות או כלים and says the chiyuv is only 1/2 nezek, as Rashi explains - due to the fact that this is meshune. Does this automatically mean we have jumped to keren? Perhaps not. My Rebbe, Rav Lichtenstein שליט"א points out that the Rambam brings this halacha in perek 3 of hilchos nizkei mammon, which discusses nizkei shein. If it were really supposed to be keren, then it would have made sense for this to be elsewhere in the Rambam's discussion of halachos related to keren. 

This raises a possibility that meshune is a general din, not necessarily related to keren per se but any av. This would work well with the first mishna in the second perek as well which changes from regular regel to mevaetes (see tosfos there who says we did jump to keren but it is brought here since the din is parallel). There is one problem though with applying meshune to regel since they seem to be contradictory as the definition of regel is normative - היזיקו מצוי. Rav Lichtenstein suggested that the בעיטה that is meshune is in the context of a normal action of walking, as opposed to בעיטה we saw earlier that is pure keren -- in mishna 15b.

This solves another problem in that we we have been seeing two different chiyuvim of keren: כוונתו להזיק and משונה and they dont necessarily go together. So, we would now say that only כוונתו להזיק is keren. Of course we need to work this out with the issue if petur reshus harabim as well.

בתר מעיקרא או בתר תבר מנא אזלינן Bava Kama 17b

Rashi explains the din of Rabba where a person threw an item and then someone else broke it, that patur mean that the thrower is chayav and thus we see that Rabba holds בתר מעיקרא. The Ramban explains that the Rif understood Rabba differently, that Rabba actually holds בתר תבר מנא אזלינן and the meaning of patur is that therefore no one is chayav, since the thrower did not break it and the person who actually broke it broke a worthless object since it was about to be broken.

Rav Soloveitchik (רשימות שיעורים) explained this machlokes between Ramban and Rashi as follows:
According to the Ramban, each perpetrator should be looked at inidividually. For this reason, there is no chiyuv. The one who threw it would have been chayav if we held בתר מעיקרא and even had he died before the nezek happened, would have been chayav. But since we hold בתר תבר מנא, the only significant time is the breakage when the item was worthless. However, Rashi holds that we need to consider both times -- מעיקרא and תבר מנא, as both together caused the damage. The question of the sugya is who do we pin the chiyuv on (not who is responsible). Whichever we pick will create a chiyuv on the one individual and uproot the other's chiyuv, and creates an obligation on the entire act (thus having to pay even if we say בתר תבר מנא).

One other interesting note on this. The Ramban asked on Rashi from a case of murder: where a person throws a baby off a roof and someone else catches the baby on his sword (26b). In this case,the sugya says that the thrower is patur. What is the difference? Rav Soloveitchik points out the Ramban's question is very difficult. Unlike an inanimate object that we may consider worthless and broken as soon as it will break, this is not true in the halacha of people. As long as the baby is still alive it has a full status of being alive in the spheres of יבום and ערכין. Therefore, there is no possibility to consider בתר מעיקרא by murder.

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.