"It says my stats were not found?"

This suggests that your stats are private. You can change this in your Bungie settings. This could also suggest that you dont currently have a destiny membership or have any characters at the moment.

"It isn't showing a raid/dungeon badge when i got it?"

Firstly, all our badges are from FRESH ONLY. Furthermore, EVERYONE needs to have started from fresh, if anyone joins mid raid/dungeon then the badge wont count (this also means if you switch characters throughout the activity it will also NOT count).

"Why does it say 'Master Completions' for Leviathan when it should be 'Prestige Completions'?"

We chose to use the term 'Master' as it is more recognizable to the average player. We also use the term 'Master' for the other raids as well.

"Why does it say 'Contest' when I never got the contest triumph?"

We use the term 'Contest' to refer to a 'Day One' completion. We chose to use this term instead of 'Day One' as some raids had a 48 hour contest time.