The drone in the picture consolidate 3" to the gunship.
Can the gunship fight in close combat?
Can the gunship move and shoot in the next round? Only targeting the drone? Why? Based on what rule?
Yes, it can, unless it has a rule that prevents it from doing so. I don't believe there is one, but check its datasheet.
Yes, it can. It can target anything freely. AIRCRAFT don't count as making a Fall Back move when moving away from enemy models.