First, let's address the expansion of lungs, because you say "little air", but in terms of volume, lungs are very big. On average, the volume of a human body is about 65 liters. When person fully exhales, the lung capacity is at about 1-1.5L; when expanded, it's about 5-6L. Interpreted charitably, that's roughly 8% percent of the entire human body volume. So realistically, expansion of the body by 8% is the difference between slowly sinking, and floating with the top of your skull (or roughly 1% of your body volume) peaking out of water.
Now, Godzilla, on the other hand, has like 80% of his body above water. Can you imagine, the amount of expansion that needs to happen for that much buoyancy? That's pufferfish territory.
So no, a "tiny percentage" increase in body volume driven by empty chamber "inside" his body would not be enough.
Yes, but birds are very light in general. Most of their volume is feathers and they have a low bone density to boot. As the result, they have a very hard time diving, and have to either dive at high speed or paddle really hard to stay underwater.
And regarding boats, it depends. Do you mean completely empty passenger boats? Then yes, their density is very low by design, because they are mostly empty on the inside. When fully loaded, a commercial cargo vessel, is 80-90% under water.