Why does Sefer Yehoshua mention the death of Moshe at its beginning?
Rashi: This is to connect it to the Torah, which ends with the death of Moshe.
Malbim: As long as Moshe was alive, Yehoshua could not lead the army, and Hashem swore that Moshe will not cross the Yarden. Now that Moshe died, Yehoshua can lead Yisrael into Eretz Yisrael.
Was this right after Moshe's death?
Radak: It was not immediately 1 , rather, after some time.
Radak (3:2): It was right after the mourning over Moshe finished (on Nisan 7). Ru'ach ha'Kodesh does not rest on someone in mourning, like we find regarding Elisha 2 and Yakov (Bereishis 45:27) (it returned after he ceased mourning over Yosef).
Refer to Bereishis 15:1:1:1*. See Radak, who discusses why some Seforim start with "va'Yhi", and the implications of 'Panim' and 'Achor'.
Who is considered an "Eved Hashem"?
Radak: It is one who puts all his strength, intent and supervision on [Avodas] Hashem, and even when he engages in worldly matters, he intends for Avodas Hashem, like a slave [should always intend] to [serve] his master.
Malbim: If a Tzadik engages only in his own perfection, his Avdus ceases when he dies. If one seeks the perfection of the generation and coming generations, he merits and brings merit to later generations. It is called his Avodah also after death. Therefore Moshe is called Eved Hashem even after he died.
Why is he called "Yehoshua bin Nun" (with a Chirik, and not 'ben Nun')?
Refer to Shemos 33:11:7:1-6.
Why is there no Nekudah under the Aleph in "Leimor"?
Radak: The word 'Emor' gets the prefix 'Lamed' more than other prefixes, therefore the Aleph is hidden (not pronounced at all).
Why does it say "Leimor"?
Radak: He should tell Yisrael that Hashem commanded to cross the Yarden, and He will give it to us.
Why is Yehoshua called "Meshares Moshe"?