Why was the Parah given specifically to Elazar?
Rashi #1, Ramban (citing the Sifri) and Targum Yonasan: Because he was the Segen (deputy) Kohen Gadol, and the Mitzvah 1 was incumbent upon the Segen.
Rashi #2 (citing R. Moshe ha'Darshan): Just as they assembled before Aharon to make the Eigel - only Aharon was not eligible to arrange the Parah. 2
What are the connotations of "mi'Chutz la'Machaneh"?
Rashi #1: It means that the cow must be taken outside all three camps (Machaneh Kehunah, Machaneh Leviyah and Machaneh Yisrael).
And what are the implcations of "ve'Shachat osah Lefanav"?
Rashi #1 and Targum Yonasan: It implies that a Zar (another Kohen - Targum Yonasan) should Shecht it in the presence of Elazar.