He didn't 'spiral into the ground', he was able to land his machine behind the Allied lines, his last words as recorded by the first troops to reach him were something like 'is kaput'. No small achievement given the nature of his wound.
Souvenir hunters subsequently stripped him and his aircraft of everything until there was only a heap of wreckage and his naked body left. There are a number of items in museums in Australia I believe such as his flying boots. What was left was recovered by the RAF to Bertangles airfield. Richthofen was buried in the churchyard there with full military honours but removed after the armistice at the request of the locals to the German cemetery at, I think, Fricourts. He was later moved again back to Germany, to Berlin, before finally being laid to rest in his family plot in East Prussia.
As to exactly who shot him down the truth will never be known but Brown was closer than the gunner on the ground & had twin machine guns so I would give the kill to the RAF over the army!!