Why did he go back?
Radak: It was to kiss his parents, even though the verse did not explicitly say so.
Why did he use Klei ha'Bakar to cook the meat?
Radak: He was in such a rush to follow Eliyahu that he did not delay to chop firewood. Rather, he used Klei ha'Bakar (plowing Kelim) to cook it.
What is the meaning of "Bishlam ha'Basar"?
Rashi citing Targum Yonasan, Dunash and Menachem: Bishel Lahem (he cooked the meat for them. They explain bi'Shlam to be two words. Radak - the suffix Mem is in place of 'Lahem'. Also "Eretz ha'Negev Nesatani" (Yehoshua 15:19) is like Nasata Li. Here, the Mem is a pronoun for the Am, even though it was not mentioned yet, just like "va'Tir'ehu Es ha'Yeled" (Shemos 2:6) and "v'Zihamaso Chayaso Lachem" (Iyov 33:20).
Rashi: We can say that it is one word. (The suffix Mem is in place of 'Osam'.) He cooked them, i.e. the oxen in a Tzemed - there are always at least two - [he cooked] their meat.
Who are the Am to whom he gave meat?
Radak: There were only 11 plowing with him in the field! Rather, when he went to ask permission from his parents, people of his city came to accompany him, for he was about to separate from them.
Why did he give meat to everyone?
Rashi: He made a feast amidst his great Simchah.