The special effects crew failed to rotoscope King Ghidorah's rays when the huge bridge is destroyed. As the shot now plays, the bridge is destroyed by a series of explosions of unknown origin when King Ghidorah simply flies over it.
On the US Naval ship in 1944, the Captain refers twice to another officer as, "Major (Spielberg)". The rank of Major doesn't exist in the US Navy, and the collar device Spielberg wore was that of Lt. Commander.
When Ghidorah is jumping on a downed Godzilla during the fight scene, you can clearly see that Ghidorah's foot bends as he lands on Godzilla. This shows that he was made of some sort of rubber (which also explains the lack of leg movement as he jumps.)