"I'm in trouble! I lost my keys."
かぎ can mean both key and lock, so sometimes in writing, カギ is used to specifically mean "key". Use is not consistent. But I have not seen カギ used for lock.
Why is こまりました not accepted? Just an arbitrary DL choice, or would you not say that in this situation?
I don't understand the translation of たいへんです to "I'm in trouble". Is it more like "i have a problem? Is there kanji for this phase?
from what I understand, one of the uses of たいへん (大変) can be to express a bad situation, like it's being used here. I'm not sure what the exact translation would be, but I'm also not entirely sure "I'm in trouble" is the closest to the actual meaning. In this context I'm pretty sure it's just expressing something along the lines of "this is bad." (feel free to correct me if I'm wrong on any of this.)
Not an expert either but it seems more like an "Oh, no!" here than anything else
When I clicked on "lost", it suggested をなくした, but when I actually put that, it was counted as wrong?
The verb なくす conjugates to なくした in its plain form, while なくしました means the exact same thing except it's the polite form. They really need to have either be acceptable.
Why is です necessary after たいへん? Isn't that just an expression of politeness? What if I wanted to say this informally?
です is used because it's supposed to be polite in this case. You can just say たいへん to say it informally
Why nakushita is not correct? I thought it could either be nakushimashita or nakushita
なくした is much more casual than なくしました. Considering the first half of the question uses です, it would be very strange to suddenly change the level of politeness