We're also talking about a movie where a 60-foot monster is tearing the shit out of New York City. You have to be able to suspend your belief to have fun with it. Plus, look at all the crap the camera went through and still filmed perfectly fine. Who's to say it couldn't survive a nuclear bomb?
If they were going to hit it with a nuclear or traditional bomb that would "level the city," the chances of that camera surviving would be next to zero anyway.
And why would the military play the warning sirens if all they were going to do is drop some really big bombs? Did you notice all the firepower they were hitting it with up to that point, and they didn't use any giant voice signals.
3. Although the soldier refers to "level[ing] Manhattan," being "prepared to let this whole area go," and "Operation Hammer Down," no one uses the phrase "nuclear weapon." The military may have been prepared to go that far, but there is no evidence that they use nuclear weapons during the events depicted the film. Additionally even if the military had been prepared to use such weapons they do not control the use of nuclear weapons on the ground. It would have required direct Presidential action to authorize the use of a nuclear weapon.
Also, in the third reason that you quoted before, you have to realize that the military uses a lot of slang words and other jargon, so just because they don't outright say they're going to drop a nuclear bomb, doesn't mean that wasn't what they had planned to do.
Plus, the on-scene commander would most certainly know the plans to level the city, especially if it were a nuclear bomb. Sure, the order would come down from the president, but you'd be silly not to think that the commanders on the ground wouldn't be in constant communication with everyone from the chairman of the joint chiefs of staff to the president.
This is something we can probably all argue about for a long time, but it might be something we never find out for sure.