I just caught an emperor butterfly today! This is my second day of playing. I’m sure you’ll run into them eventually.
I have yet to find the tuna and koi but soon hopefully.
I've seen and caught a horned dynastid. I think the red snapper is supposed to be rare too and I caught one. I still haven't seen the ones you've listed, either. I also saw someone with a Puffer fish in their Market Box.
By "time-sensitive" do you mean if they show up at different times of day and seasons like in ACNL? I don't think so but staying in one area too long makes the number of fish and bugs that spawn decrease for a while.