When did Yehoshua send the spies?
Rashi: It was during the mourning over Moshe, for Bnei Yisrael cross the Yarden three days after the Aveilus finished, and the spies waited three days (for those chasing them to return) until they returned to Yehoshua on the fourth day, and Yehoshua crossed on the next day.
Malbim: It was when he spoke to Bnei Gad and Bnei Reuven, before Hashem spoke to him, and before he told them to prepare Tzedah.
What is the meaning of "Cheresh", and why should they go like this?
Rashi #1 (from Targum Yonasan): It is Berez - act like deaf people, so they will not hide their matters from you.
Rashi #2: It is like Cheres (earthenware). Load yourselves with pots, so it will look like you are potters.
Radak: They should go quietly, like deaf-mutes, so people will not notice them.
Radak (from Targum Yonasan), Malbim: It is Berez - covertly, so all Yisrael will not know, lest they fear sending spies. Yehoshua sent them only because he knew that they will succeed, and encourage the hearts of Bnei Yisrael with good tidings.
Why did he mention Yericho? It is part of ha'Aretz!
Rashi: It was as hard [to capture] as the rest of the land 1 , for it was on the border (so it was greatly fortified).
Radak: It was a great city.
Malbim: The Shelichus was primarily about Yericho and the land near it.
What is the meaning of "Zonah"?
Rashi (from Targum Yonasan): She was an innkeeper 1 , and sold kinds of Mezonos (food).
Radak, Malbim: A harlot. Also Targum Yonasan means this; he often translates harlot as Pundakisa, e.g. "Shetayim Nashim Zonos" (Melachim 1 3:16). Even though sometimes he translates it "Nafkas Bera", this is because a harlot is like an innkeeper; she makes herself Hefker to all (anyone may lodge with her). 2
Lev Yitzchak: Targum Yerushalmi (it seems that this should say 'Yonason') on Bereishis 42:6 (based on Bereishis Rabah 91:6) supports this, and Targum Yonasan on Bamidbar 24:14. (The latter can mean one who sells food! He explains that Bil'am counseled to set up Pundekin, and put in them lewd women selling food and drink for a discount
Why does it say "Shenayim Anashim"?
Radak: They were good men, unlike those that went on Moshe's Shelichus. 1
Also Moshe's Sheluchim were initially Tzadikim! Perhaps this is why he said 'went on Moshe's Shelichus' and not 'that Moshe sent', for they were Tzadikim when he sent them, but not when they did the Shelichus. (PF)
Why does it say "va'Yishkevu Shamah"?
Radak: This teaches that they were there only one night.
After seeing the calamity that came from Moshe's spies, why did Yehoshua send spies?
Malbim: There were five differences between them. (a) The people requested that Moshe send spies, therefore they relied on their report. Here, Yehoshua chose to send them. (b) Moshe sent them from Midbar Paran, far from the land, and they were unsure about the essence of the land and their ability to conquer it. Yehoshua sent from Shitim, on the land's border. There was no doubt about entering, only about the easiest place to conquer. (c) Moshe sent 12 Tarim, and Yehoshua sent two Meraglim (spies). A Tar seeks the good parts of the land and whether the people are strong; spies seek Ervas ha'Aretz
Some explain simply, that Zonah is a harlot. Why would the spies go to a harlot?!
Malbim #1: This is lest Kena'anim know that they are from Bnei Yisrael. They knew that Am Yisrael hates Zenus. It would not cross their minds that Sheluchim of Yisrael would go to a harlot!
Malbim #2: Rachav was famous among the important Kena'anim, so she knew their secrets, and they could ask her about all happenings in the land. 1
It was Hashgachah that they went to her; only she merited to help them and save her family. Being the greatest harlot, she knew best that no man had any strength left! (PF)
Did the spies ask people who is the greatest harlot? Perhaps they knew through Ru'ach ha'Kodesh. (PF)