Emerald - First off, I'm sorry for your loss. I know the sorrow of losing a little soul like that.
Secondly, I highly doubt his death had anything to do with dirty water or food. It sounds as if he had been ill for quite a long time, I'm guessing probably since the beginning of June when you started thinking he was "depressed." I'd guess he somehow succumbed to some kind of bacterial infection. It's hard to tell what kind without cultures, because for the most part, the symptoms are all the same - a fluffy, mostly sedentary bird acting like it doesn't feel good for some reason. By the time he stopped perching and was sitting on the bottom of his cage, it was far too late. Even if you'd rushed him to the vet at that point, it's unlikely they'd have been able to do much for him.
To answer your original question: Do birds get depressed? Yes, they can. That said, I've never heard of it leading to death in the way you described, and I've never heard of a budgie succumbing to depression the way big birds tend to do - they tend to be emotionally stronger than many of the big birds - that's another plus for our little guys.
Additionally, it's highly doubtful that he was stressed to death either. Though that's a more likely scenario than him being depressed, over the years we've learned that the little dudes were far hardier than we used to think.
So if you see fit, you can let your little girl off the hook on that one! lol!
Whenever you have an unexplained death, it's always a good idea to disinfect the cage they were in. I used a generous amount of bleach and blue Dawn dish soap in a big bucket of water and clean clean clean. Then I rinse rinse rinse and let it dry in the sun. I figure nothing is going to live through that.
BlueBelle will be fine, if she's not already, and I wouldn't hesitate to get another budgie when you feel the time is right. Just remember when you do that you'll need to quarantine him for 30 days in a separate room. This will insure that He doesn't introduce anything harmful to your Bluebelle. Further, it's not uncommon for any bird to become ill following the stress of moving to a new place. It lowers their immune system and allows an illness they'd usually be able to fight, to come forth and rear it's ugly head. If the bird is in quarantine, you'll be watching it closely and can seek treatment as soon as you notice something is "off."
I hope all of this has helped some. Take time to heal and and enjoy your Bluebelle, then move forward. It's highly unlikely that y'all had anything to do with losing your sweet Emerald.