How did the piece of wood turn the bitter water sweet?
Ramban #1: Hashem taught Moshe that particular species of wood had in its nature to sweeten bitter water. 1
Ramban #2 (citing the Mechilta, Midrash Tanchuma): The wood, like the water, was bitter, and the fact that it turned the water sweet was a miracle within a miracle. 2
Targum Yonasan: Moshe wrote the Great Name of Hashem (Havayah) on it.
What is the significance of the word "Vayorehu Hashem Eitz". What did Hashem teach Moshe?"
Ramban #3 (citing the Tanchuma): Hashem taught Moshe His ways - that He sweetens the bitter with things that are bitter.
What do "Chok" and "Mishpat" refer to?
Ramban #2: "Chok" and "Mishpat" 3 both refer to the customs and regular practices that Hashem introduced when they entered the great and awesome desert, customs to busy themselves with until they entered an inhabited land.
Ramban #3: With reference to "ve'Sham Nisahu", Hashem chastised them by making them suffer Chok - the law of the desert, to suffer hunger and thirst, thereby inducing them to cry out to Him (not by complaining), and Mishpat - He taught them a series of civil laws by which to live by - to learn to love one another, to obey the instructions of their leaders, laws of Tzeni'us regarding their wives and children; how to behave with respect towards peddlers who entered the camp to sell their wares, and how to behave ethically when plundering their enemies - not to indulge in the abominations of the other nations. 4
Da'as Zekenim: It is a Chok that is a Mishpat, i.e. Kivud Av v'Em.
Targum Yonasan switches Kibud Av va'Eim for Parah Adumah. Rashi in Devarim (5:16) includes Kibud va'Em. See Sifsei Chachamim.
See Targum Yonasan, who lists the areas of Dinim that Hashem taught them.
The Ramban cites many cases where that is what "Chok" and "Mishpat" means (See for example Yirmeyah, 33:25 regarding "Chok", and Bereishis 40:13 regarding "Mishpat", and Yehoshua, 24:25 [Ramban]. See also Ba'al ha'Turim).
What test is the Pasuk referring to?
Rashi: Hashem tested them by depriving them of water and watching their reaction. They failed by grumbling, instead of asking Moshe nicely to pray on their behalf. 1
Ramban #1 (interpreting Rashi's statement DH 'Sham Sam lo'): He introduced them to the aforementioned Mitzvos (in advance of Matan Torah) 2 to see whether they would accept them willingly and with joy, or reluctantly. 3
Ramban #2: Refer to 15:25:3:3.
Seforno: He tested Yisrael to see whether they would receive the Chok and Mishpat that He now presented them with, or whether they would reject them and revert to their old ways, because provided that they would accept them and continue to observe them, they would not suffer the plagues of the Egyptians. 4
Targum Yonasan: Yisrael tried Hashem with ten tests. 5
See Ramban's objection to this explanation.
Just as Avraham Avinu performed the Mitzvos before they were commanded. It also trained them in the performance of Mitzvos (Ramban).
And the following Pasuk informed them that there were more Mitzvos to come (Ramban).
As the following Pasuk spells out (Seforno).
See Bartenura, Avos 5:4, and Na'ar Yonasan.
Rashi writes that they were commanded about Shabbos. Below (16:22), Rashi writes that they were surprised to find double manna on Friday, for they had not yet been told about Shabbos!
Moshav Zekenim: This is a Drashah, and we do not challenge a Drashah. 1
Riva (16:1): Hashem told Moshe to tell Yisrael about Shabbos at Marah, but Moshe delayed telling them, like he delayed 2 telling them that extra Manna would fall on Erev Shabbos (refer to 16:22:2:1). Hashem punished (rebuked) him
Seemingly, the two Drashos contradict each other. How can we resolve this Drashah with the verses below, that the Nesi'im told Moshe that they collected double? Perhaps they expected that they will receive a single portion, and miraculously, it will suffice for two days (like the dough that they took from Mitzrayim
Rashi writes that they were commanded about Parah Adumah. What is the source for this? Sanhedrin 56b lists 10 Mitzvos that they were commanded, and Parah Adumah is not among them!
Moshav Zekenim citing Bechor Shor: Moshe asked how bitter wood can sweeten water 1 , and Hashem said that there is a Mitzvah like this - Parah Adumah is Metamei Tehorim (and Metaher Temei
Terumas ha'Deshen: Since Yisrael saw this, they will be able to accept that Parah is Metamei Tehorim and vice-versa. If not, there was no need to teach this to them here!