1) If someone is lost in the desert and doesn't know when Shabbos is he/she has to make kiddush and havdalah one day a week as if that day was Shabbos, and on any given day no melacha except for what is needed for pikuach nefesh may be done lest that day is Shabbos. The MG"A asks why this should be so. Shabbos is only one day out of seven. Whatever day is Shabbos should be bateil b'rov to the six days of chol.
Question: how can you speak of bitul brov (or kavua) with respect to days? The chiyuv to keeps Shabbos is a chovas ha'gavra (see R' Yosef Engel in Esvan D'Oraysa re: whether time bound chiyuvim are issurei gavra or issurei cheftza) on the person, not a chiyuv of the day. The person lost in the desert, the gavra, is not bateil to anything?
When I saw this question I first thought it was great and now I'm not sure it makes sense. True, keeping Shabbos may be an issur gavra, but you have to define the day before you can say the chovas ha'gavra gets off the ground. It's an intrinsic condition to the chiyuv.
2) Mah tovu ohalecha Ya'akov. Rashi explains that Bilam saw that ain pischeihem mechuvanim.
Chazal teach us that if we just open a pesach k'chudo shel machat, an opening the size of the hole in a needle, to let Hashem into our hearts, he will open for us a pesach as wide as the door of the heichal. R' Meir Shapiro explained that this is what Bilam saw. Ain pischeihem mechuvanim: the door Hashem opens for us is completely out of proportion to the door we opened for him -- the two are not aligned. But Hashem loves us, so that's the way it is.
3) The gemara (A"Z 4) writes that Bilam's success was due to the fact that he was able to figure out when the one moment of the day that Hashem gets angry. Hashem did a miracle and withheld his anger the entire time that Bilam tried to curse us.
The gemara continues that R' Yehoshua ben Levi had an obnoxious neighbor who was a min and drove him crazy, so he decided to wait for that moment of Hashem's anger and then ask Hashem to do away with this neighbor. The moment came, but just then RYb"L fell asleep. He took this as a sign that v'rachamav al kol ma'asav, Hashem has mercy even on the wicked and did not like his plan.
If Hashem gets angry for this one moment every single day, there must be some need in the seder of the world for such a thing to happen. So why withhold that anger just to thwart Bilam? Hashem, for example, does not stop the sun from rising just because idolaters worship it. Why didn't Hashem just make Bilam fall asleep like he did to RYb"L instead of interrupting the course of nature?
We've discussed lots of times (e.g. here, here, and other places ) the famous view of the Ohr haChaim (and others) that while Hashem can force animals and inanimate objects to conform to his plan, a ba'al bechira, a human being that has free choice, has far more latitude and can do an end run around Hashem's designs. What that (probably) means is not that Hashem does not have control over people -- what it means is that it takes for more zechuyos to cause/ask for Hashem to interfere with a ba'al bechira.
Of course if Bilam just fell asleep his plan would have been thwarted. Our parsha is telling us a bigger chiddush, explains R' Yerucham Lebovitz. Even though Bilam was awake and had free choice as a ba'al bechira to act against Klal Yisrael, he still did not succeed.
4) V'lo ra'ah amal b'Yisrael... A beautiful Ohr haChaim here
גם נתכוון לומר שהצדיקים הגם שעושים מצות וכל עסקם בתורה אינם מרגישים שיש להם עמל, על דרך אומרו (תהלים עג) עמל הוא בעיני אלא אדרבא כאדם המרויח וכאדם המשתעשע בשעשועים לרוב חשקם בתורה
Mitzvos should not been seen as a burden or bother -- amal -- but rather as a pleasure to do.
R' Ya'akov Neiman in his Darkei Musar quotes a famous mashal (I'll write it over anyway : )of the Dubno Magid which the Kotzker said must have been given b'ruach hakodesh. V'lo oso karasa Ya'akov, ki yagata bi Yisrael (Yeshaya 43:22). Hashem criticizes Klal Yisrael for not calling to him, for being weary of him. The mashal: the was a royal officer who was travelling through some town, and when he got off the train he went ahead to his hotel and left his bags to be brought later. Later that day the bellhop, huffing and puffing and sweating from the exertion, knocked on the hotel door and told him that he had brought the suitcases. Without even looking, the officer replied that he was confused and had brought someone else's bags. "How do you know?" the bellhop asked. "You didn't even look at them!' "Because," answered the officer, "My bags were light -- you obviously have been struggling with whatever you brought, and so I know they are not mine."
Hashem tells Klal Yisrael, "OSI lo karasa," whatever frumkeit you have been killing yourself over and struggling with, it's not MY frumkeit, that's not MY Torah and mitzvos, "ki yagata bi Yisrael," because whatever it is you think you are doing is an unbearable and painful burden. Those suitcases you've been struggling with, says Hashem, are not my suitcases. When you are schlepping my suitcases, "v'lo ra'ah amal b'Yisrael," they are no bother at all.