What is the meaning of "Nitzav Al Avdei Sha'ul"?
Radak, Targum Yonasan: He stood at their head. He was the greatest of Sha'ul's servants ha'Nitzavim Alav. 'Nitzav' refers to opposite matters - Nitzavim Alav is in front of him and near him, to do what he will command. "Nitzav Al Avdei Sha'ul" means that he was the head, and appointed over them. This is like "vNitzav Al ha'Kotzrim" (Rus 2:5).
Did Do'eg accurately say what he saw?
Malbim (9:10): No. Had he done so, no harm would have come. David came to Beis Hashem, not to Beis Achimelech! Do'eg omitted that David said that he was doing Sha'ul's Shelichus, 1 and Bulmus had seized him, and he asked from the Urim v'Tumim Stam (without telling the question to Achimelech), and he asked Stam for a sword (not specifically Galyus'), just there was no other sword. Also refer to 22:9:3:1, 22:10:1:1, 22:10:2:1,
Malbim: Since it was urgent to go promptly, he did not take a sword and food. (Did he not realize from the beginning that he will need them? Perhaps he needed to go immediately to another place before Beis Hashem, and expected to get them there. Also refer to 22:10:1:1***. - PF)
Why did he call him Achimelech ben Achituv, and not Achimelech ha'Kohen?
Malbim: He did not go to him because he was the Kohen (i.e. in order to ask from the Urim v'Tumim - PF), rather, because he was Ben Achituv; his family is allied with David. It says "va'Yaged l'Sha'ul Ba David El Beis Achimelech" (Tehilim 52:2), for this was the basis of the slander. Sha'ul could think that he already told him the question in his house. Rather, Do'eg saw that David went directly to Beis Hashem, and never told Achimelech his question. 1
I.e. he is not worthy to be called by his own name. Refer to 20:30:3:1* (PF).