Your results are sooooooo confusing...
Try adding a bufer so that the pH value is about 5 (acetic bufer) and add soduim sulphate. If a white precipitate forms, then you really have barium cation.
Now, about the anion: What anion can you get? Any?
It may be a posphate, so it would be possible that they gave you
Ba(NH4)PO4. Could it?
Chlorides can give the same smell as acetic anion if they are in large concentration.
The reaction with AgNO3/NH3/HNO3 giving the white precipitate, can be Cl- but could be carbonates, too. When you dissolve your solid substance in strong acid, what happens? Can you see large bubbles?
Oh, and anions CAN NOT give flame tests. They can't color the flame!