Who were the "Moshlim"?
Rashi (citing Midrash Rabah): They were 'the parable-sayers? with reference to Bil'am and his father Be'or. 1
Ramban: Refer to 21:14:2:2*.
Seforno: They were people who interpreted the parables that they dreamt, such as Bil'am. 2
Rashbam: They were people who predicted the future, such as Bil'am and his contemporaries.
Targum Yonasan and Bava Basra 78b: They were Tzadikim, who rule over their Yeitzer ha'Ra. 3
Moshav Zekenim: They were officers whom Sichon appointed over the cities of Mo'av that he conquered.
What is "Al-Kein" referring to?
Rashi and Rashbam: It refers to the battle between Sichon and Mo'av.
To whom did the Moshlim say 'Bo'u Cheshbon!'?
Rashi and Ramban: They cursed Mo'av 1 that Cheshbon should fall into the hands of Sichon, 2 and that it should be rebuilt as the capital of Sichon.
Seforno: They were telling the inhabitants of Ar, the capital of Mo'av, to come to Cheshbon, the capital of Sichon and to submit to his sovereignty, 3 because he was destined to defeat them.
Targum Yonasan and Bava Basra 78b: Those who rule over their Yeitzer ha'Ra 4 should say to themselves 'Come, let us make a calculation - the gains of an Aveirah against the punishment, and the losses of a Mitzvah against the reward'. 5
Da'as Zekenim and Hadar Zekenim: He was telling people who were afraid to live there whilst the king of Mo'av held it - because the mighty Sichon laid claim to it, that, now that he had conquered it, they should come and live there, since there was no longer any reason to be afraid.
Ramban: They predicted that Sichon would come and destroy Cheshbon and subsequently rebuild it as his own capital.
Rashi: Sichon hired Bil'am to curse it after failing to defeat Cheshbon. And it is due to that curse that Balak later tells Bil'am - in 22:6 - "for I know that whoever you bless is blessed, and whoever you curse is cursed."
From the fact that the Rashbam refers to Ar as the capital of Mo'av, he seems to explain the Pasuk like the Seforno.
Mesilas Yesharim (Perek 3): Only one who has ceased to be under his Yeitzer ha'Ra, and rules over it, can give this true counsel One who is still under its influence, his eyes do not see this truth and does not recognize it, for his Yeitzer literally blinds his eyes.
Targum Yonason: The one who awakens and speaks words of Torah will be established!' (We read "Ir" like 'Er' - awake, and "Sichon", like Sichah - talk.)
What are the connotations of "Tibaneh ve'Sikonein Ir Sichon!"?
Rashi and Seforno: It means 'Cheshbon will be built and established in the name of Sichon as his capitol'.
Targum Yonasan: 'The one who is aroused and who speaks words of Torah will be built-up and established.
Why were Yisrael not commanded to Kasher the vessels that they captured here, like they were commanded to after the war against Midyan in 31:21-23?
Moshav Zekenim citing Bechor Shor: Because this was part of the conquest of Cana'an, where Even pig-flanks were permitted, 1 as the Torah writes in Va'eschanan Devarim 6:11 - "u'Vatim Mele'im Kol Tuv", and it was therefore not necessary to Kasher the Keilim. The war against Midyan, on the other hand, was a war of vengeance with no connection to the conquest of Cana'an.
Moshav Zekenim: See Chulin 17a.