bris bein ha'besarim -- nevuah or havtacha

1. In R' Shternbruch's teshuvos (vol 1) quotes an interesting sevara from R' Shmuel Rozovsky: since an aveil is exempt from leaning Torah and even kri'as haTorah (during the week if an aveil is called up for an aliya he has to decline and not go up), therefore the aveil is not mitztareif to a minyan for krias haTorah.  Meaning, if there are 9 people in the minyan + the aveil, there is not a tzibur that is chayav in kri'as haTorah.  R' Shternbruch does not pasken this way, but it's an interesting idea.

2. The first Rashi in chumash that says the Torah opens with the idea of creation to prove that all the land in the world belongs to G-d and he can therefore give us "nachalas goyim."  My BIL R' Yochanan in his parsha thought for Braishis here commented on this expression of "nachalas goyim."  "Why is it called an inheritance of nations, implying that Israel rightfully belonged to them? It belongs to us!"

Not that he needs my help, but I would strengthen the question.  The Rambam in Sefer haMitzvos (third shoresh) writes that the command Moshe was given not to attack Eisav who was living in Se'ir, and the command not to attack Amon and Moav are not counted as mitzvos.  A mitzvah, says the Rambam, has to apply for all eternity.  One day Moshiach will come and we will have permission and be zocheh to conquer all of these lands.  Ramban disagrees.  These lands, says Ramban, are called a "yerusha," and there is a principle in halacha that nachala ain lah hefesk, an inheritance is forever, it passes from generation to generation automatically.  The land of Eisav, Amon, and Moav are forever off limits for us.  So how then will be allowed to one day conquer this territory?  Ramban answers that Sancherev mixed up all the people in the world.  Amon and Moav are no longer living in their ancestral home, Eisav is not living in his ancestral homeland, and so when we conquer these lands it is really a different people that we will one day be conquering.   

We see from Ramban that when something is given as a yerusha = nachala, it is forever.  So why then does the pasuk speak of the land of Israel as "nachalas goyim" when the land was meant ultimately for us?! 

You have to read his answer there.

3. Now that we know this Ramban, when we come to the promise of Bris ben hBesarim in our parsha of the land of 10 nations being given to Avraham, we have to read a little more carefully.  One of those 10 lands is the Se'ir land of Eisav, and two of the lands are the territory of Amon and Moav, given to Lot's decedents.  If it belongs to them forever -- nachala ain lah hefsek -- how then could G-d promise the same land to Avraham? 

It must be (see Avi Ezri Hil Melachim 6:1) that Avraham was not actually being given the land directly, but was being shown prophetically that his children will eventually conquer those lands after Sancherev does his population resettlement.  It is a nevuah, not a havtacha.  The Rambam disagrees with this whole theory and l'shitaso holds (Rotzeiach 8:4) that the 10 lands were promised to Avraham directly and never given as a permanent inheritance to anyone else.  

4. "V'lo nasa osam ha'aretz la'sheves yachdav ki haya rechusham rav v'lo yachlu lasheves yachdav."  (13:6)  First the pasuk tells us that Avraham and Lot could not live together because there was not enough grazing land for all their flocks, but then the pasuk again repeats, "v'lo yachlu lasheves yachdav," that they could not live together.  Why the redundancy?

We see from here, explains R' Shimon Sofer, how machlokes works.  At first, there might be a very good reason for disagreement.  Avraham and Lot and their shepherds could not agree on who should graze where.  But at some point, once the fire of machlokes is lit, even if the reason has long since been forgotten or ameliorated, the fight continues of its own accord.  "V'lo yachlu lasheves yachdav" now irrespective of the reason the fight started.  

Once started, machlokes will take on a life of its own.