I don't think the app shows the order of the bosses that you have beaten in the raid, just the number of bosses you have beaten. I once joined some friends for a Crota raid after they had beaten the thrall maze (I joined them on the bridge) and we finished the rest of the raid. However, after the raid the app showed that I had beaten the first 3 bosses and not the last one. Then I went back on my own and did the thrall maze to get the rewards and then the app said I had beaten the "last boss" according to the order of bosses in the app even though it's the first one.
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Yea you have to do all 4 parts to get full completion on app advisor . If not it will just full up 1 at a time in no order