(Some of this is drawn from https://www.halachipedia.com/index.php?title=Animals_on_Shabbat)
I think Sefaria's language of "Sabbath-observant animal" was a bit confusing.
Any animal owned by a Jew may not be prompted to do activity prohibited for a Jew themselves to do on Sabbath.
Let's break that statement down, and hopefully all your questions ...
David's seeing Batsheva bathing was accidental and unintentional - the story is detailed in Sanhedrin 107a:
Bathsheba was shampooing her head behind a screen, which concealed
her from sight. Satan came and appeared to David as a bird. David shot
an arrow at the bird, the arrow broke the screen, Bathsheba was
exposed, and David saw her.