Did Do'eg tell what happened in the correct order?
Malbim: No. Really, he asked from the Urim v'Tumim after he gave the sword and food. 1 This implies that Achimelech did not know anything - if not, why did he give to him the sword and food in Beis Hashem, in front of Do'eg, and not covertly in his house 2 ? Do'eg said that ha asked first. After he needed to ask from the Urim v'Tumim in front of Do'eg, also the sword and food he did not strive to give it covertly. 3 Also, even after David asked, i.e. about fleeing, and Achimelech knew this 4 , he gave to him food and a sword!
What is Malbim's source? The verses above did not explicitly discuss asking, just Rashi and Targum Yonasan (21:10) say that he told David about the sword "Acharei ha'Efod", i.e. after asking from the Urim v'Tumim! (PF)
The sword was in Beis Hashem, and perhaps also the Kodesh food. Malbim means, he would have taken them to his house, and covertly given them to David there. (PF)
Also, if he knew that he will need a sword before hearing the Urim v'Tumim's answer, why did Sha'ul not send him with one?! However, in any case, why would David not take enough food? Perhaps David said that he rushed to leave, and assumed that the youths took their own rations. When he found that they did not, David gave to them his own, and traveled quickly without eating, and amidst this Bulmus seized him. (PF)
Malbim said (verse 9) that David did not tell the question to Achimelech. How could Achimelech know from the answer that David is fleeing? Perhaps David asked to where he should go, and the answer was 'Pelishtim'! (PF)
Why did he say "v'Tzeidah Nasan Lo"?
Malbim: He did not say that Bulmus had seized him, and even if he knew that David flees, he must feed him. He may not cause his death. Rather, Do'eg implied that Achimelech gave to him provisions for many days.
