Why does the Torah write here "Al Tisgar bam Milchamah", and by the B'nei Amon (in Pasuk 19) just "Al Tisgar bam"?
Rashi: Because here Hashem only forbade fighting Mo'av, but not harassing them, 1 whereas even harassing Amon was strictly forbidden.
Why were Yisrael not allowed to harass Amon at all, which was not the case with Mo'av?
Rashi: This was a reward for the modesty of their mother - the younger daughter of Lot - who did not publicize the fact that her son Amon was born from her father like her sister did by calling her son 'Mo'av' (me'Av).
What is "Ar"?
Why would Moshe think to fight Mo'av without permission, that Hashem needed to forbid this?
Bava Kama 38a: Hashem said to take vengeance against Midyan, and Moshe thought that all the more so, against Mo'av. 1
Since the Torah forbids fighting Mo'av, why did Elisha command to do so (Melachim 2:3:18,19)?
Moshav Zekenim (20:19): After Sichon took their land, it was permitted to Yisrael (Gitin 38a, and we fought only in the part that Sichon had taken).
QUESTIONS ON RASHI
Rashi writes that Hashem forbade only fighting Mo'av; Yisrael appeared armed in front of them and took spoils, and therefore Mo'av feared them. I.e. Hashem forbade fighting them before the episode with Midyan. Really, Hashem forbade this after He said to take vengeance against Midyan, and Moshe thought that all the more so, to fight Mo'av (Bava Kama 38a)!
Refer to Bamidbar 22:3:151:1,2 and the note there.