Why does the Torah see fit to write 'ha'Motz'im oso Mekoshesh"?
Rashi: This implies that even after they found him collecting wood and warned him to desist, he continued to sin. 1
See Sifsei Chachamim.
Why did he desecrate Shabbos?
Moshav Zekenim (32): He did it to be Mekadesh Shem Shamayim. Targum Yonasan (32) 1 - so that Yisrael would see the punishment for Chilul Shabbos [and not follow in his footsteps].
See Na'ar Yonasan.
Who was the Mekoshesh Eitzim?
Rashi (in Pinchas, 27:3): He was Tz'lofchad.