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" 4 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. 5
Targum Yonasan switches Kibud Av va'Eim for Parah Adumah. All the commentaries include Shabbos (See Sifsei Chachamim).
See Targum Yonasan, who lists the areas of Dinim that Hashem taught them.
Rashi in Va'eschanan (5:16) includes Kibud va'Eim in the current list.
The Ramban cites many cases where that is what "Chok" and "Mishpat" means (See for example Yirmiyah, 33:25 regarding "Chok", and Vayeshev, 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 Daven 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 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, Pirkei Avos, 5:4, and Na'ar Yonasan.