- There was a bird across the window.
This only works if the bird is splayed out across the window, or if the bird has struck the window, and its dead or injured body is spread across it. Not at all the intended meaning.
2- There was a bird across from the window.
This implies that the bird is somehow opposite the window, perhaps on the other side of a street. Again not the intended meaning.
3- There was a bird through the window.
This implies that the bird has crashed and stuck half-way through the window. Still not the intended meaning.
Possible sentence that do carry the intended meaning would be
- There was a bird on the other side of the window.
- There was a bird visible through the window.
- There was a bird beyond the window.
4- There was a little girl across the table.
Yes! This works and carries the intended meaning.
5- There was a little girl across from the table.
This says that the table is on one side of the room or space, and the little girl is on the other side. Not the intended meaning.
6- There was a little girl through the table.
Like the case of the bird, this suggests that the girl has collided with the table and gone partway through it. Clearly not the intended meaning.
Using "across" or "on the other side of" is best in thi9s case.
I don't see any simple rule for when "across" means "on the other side of". It does for "across the city" or "across the ocean". Perhaps the distinction is that those are generally thought of as horizontal things, as tables are, but windows are not. But I am not sure that this applies widely. "across from" and "through" rarely mean "on the other side of" "Across from me" may mean on the other side of a table, but not the other side of me.