1)
Why did he ask his origin? He already said that he is an Amaleki!
1.
Radak: He already said his nation. David asked if he lived in his land, i.e. Eretz Amalek, and how he came to the war. He answered that also his father came from Eretz Amalek and lived in Eretz Yisrael. Stam "Ger" refers to where one lives. Sometimes it refers to converting to the law of Yisrael. Perhaps here, his father converted.