Yes, i guess they are all correct, but under or underneath are probably the best choices.

Yes, they are all correct. Most people would have one of these two ideas: 1. The bird flew under the bridge and then back up on the other side of the bridge. 2. The bird flew under the bridge and stayed there. (a), (b), and (d) are essentially the same. In my mind, (c) can mean the bird just flew at a height lower than the bridge but the bridge was not directly over the bird. For example, the bird could have flown next to the bridge but at a height that was less than that of the bridge. However, if this is the image you want, then I would just say "The bird did not fly as high as the bridge."