Hey guys, I think I found something.
The “da” that comes and goes bothers me.
For instance, would a police person searching an apartment use “da” when telling a companion “I think I found something,” implicitly pointing to a place, whereas If I tell someone “I found something yesterday I’d been looking for” would I omit the “da” because I’m really just referring to the finding? In other words, does the presence of “da” in the cloze sentence make it implicitly referential?
Noone seems to dare attempt to explain the usage of “da”. It really is hard.
But let me say that the “da” doesn’t necessarily refer to a place. It can modify the tone, and that’s what makes it so hard to grasp. Sometimes you can think of it as a general term for that place, at that moment, or in that situation/context, but sometimes it’s not even that.
Examples:
that place:
Da lag das Buch als ich es gefunden habe.
Da hat früher mal ein Bild gehangen.
that moment:
Ich war gerade eingeschlafen, da kam ein Anruf.
Ich war noch nicht bereit, da kam schon das Startsignal.
Da hat das rote Pferd sich einfach umgekehrt und hat mit seinem Schwanz die Fliege abgewehrt. (song lyrics! 
. Maybe this also belongs into the next section…)
that situation/context:
Du hast dir den ganzen Tag den Krach anhören müssen? Ich würde da verrückt werden.
Du hast dein ganzes Geld für Süßigkeiten ausgegeben? Ich wüsste da was besseres mit dem Geld anzufangen.
Da wird doch der Hund in der Pfanne verrückt!
neither of the above?
Ich hätte da mal eine Frage…
Komm mal her, ich hab da was für dich!
Es gibt da einen gewissen Jemand, in den ich total verknallt bin.
He, Leute! Ich glaube, ich habe da was gefunden.
would a police person searching an apartment use “da” when telling a companion “I think I found something,”
Yes! Totally fine.
implicitly pointing to a place
Not necessarily, but it can.
Your overview is awesome. Thank you so much for taking the time to work through so many examples. The “da” problem raises a lot of questions about the difference between understanding and translation, which would be a good topic by itself for the forum. Achieving understanding is my goal when I hear German spoken, but it’s more like translation when I read, imagining a Cloze sentence as extracted from a story, or other text. I can’t make it any clearer, but the two modes feel different, as if I’m applying different criteria for success.