What was Miriam looking out for?
Ramban: Refer to 2:2:1:1**.
Seforno: She assumed that some Egyptian or other (who was bathing in the river) 1 would take the baby home (and she wanted to keep track of its whereabouts). 2
Sotah 13a: Miriam had prophesied that her mother would give birth to a son who would save Yisrael. When Moshe was born and the house was filled with light, her father Amram said to her ,'My daughter, your prophecy has come true!' But now, when Yocheved cast him into the Nile, he declared, 'My daughter, what happened to your prophecy?' That is why she stood by the river bank, to see how her prophecy would unfold.
In what construct is the verb "va'Teisatzav"? (If the root is Yud-Tzadi-Veis, shouldn't it be written 'va'Tis'yatzav'?)
Maharal (Gevuros Hashem Ch. 17, p. 79): The Gemara (Megilah 14a, Sotah 12b) tells us that Miriam had already prophesied that her mother would give birth to the redeemer. She now stationed herself where she could see how her prophecy would be fulfilled. "Va'Teisatzav" is then a blend of Hispa'el (reflexive) and Nif'al (passive) 1 - 'She stationed herself,' and also, the prophecy caused her to 'be stationed.'
Maharal: The vowel Tzeirei under the (first) Tav is a feature sometimes present in a Nif'al verb - e.g. in the word 'Tei'aseh' (it should be done). (But the Tzeirei in that word is caused by the letter Ayin, which cannot take the Dagesh caused by the fallen Nun of Nif'al. Arguably, in our case, 'va'Ti'tatzav' (first Tav with a Chirik, the second with a Dagesh) would accomplish the same blend of Hispa'el and Nif'al. (CS))
Why does the Pasuk use "va'Teisatzav" (root 'Yatzav'), meaning "she stationed herself," rather than simply 'va'Ta'amod' - 'she stood'?
Maharal (Gevuros Hashem Ch. 17, p. 79): Chazal (Shemos Rabah, and Sotah 11a) interpret this Pasuk as referring to the Shechinah. The word 'Yatzav' implies strength and permanence; and we find it used regarding Hashem in Shmuel I 3:10. It was Hashem who caused that no harm would befall Moshe in the river. 1
See Maharal (ibid.) further; also see Maharal (Chidushei Agados Vol. 2, p. 44, to Sotah 11a).