Why was he called "ha'Edomi"?
Radak: He lived in Eretz Edom, just like Oved Edom ha'Giti live in Gas.
Radak citing Shocher Tov 4: He is called so based on his city, which was "Nochach Ma'aleh Adumim" (Yehoshua 15:7, 18:17, which lists the borders of Yehudah and Binyamin).
Why does it say only that he came to Beis Achimelech, but not what happened there?
Radak: What happened there is known from Shmuel I, Perek 21.
Malbim: Do'eg's primary slander was saying that David came to Beis Achimelech. Really, he came to Beis Hashem. Do'eg heard that David said that he is going on a Shelichus of the king, that he is starving, and Bulmus seized him (he is in mortal danger), and he asked Stam from the Urim v'Tumim; he said that Sha'ul said to conceal the question. He asked Stam for a sword; he took Galyus', for there was no other. Do'eg changed six matters, including that he went to Beis Achimelech, and revealed that he flees Sha'ul.
