I'll answer because some people are probably genuinely curious to some of these.
> 1. What happened to the deadly radioactivity?
It mostly decayed out. Generally speaking, 8 half-lives mean that it's essentially decayed to "gone". High-level atmospheric tests usually cause it to spread out and depending on wind patterns can dissipate enough to be essentially harmless - though with precision instruments you can measure the differences throughout the whole world. Steel from shipwrecks from before the first explosion can be desirable for some of this equipment.
With explosions closer to or below ground level, there can be longer-lasting elements baked into the ground, like Trinitite (a green glass like material) that can have trace amounts of cesium-137, with a half-life of 30 years, which going by the 8 times rule means that it'll be "dangerous" for ~240 years.
Also the type of bomb matters to what is left behind. A uranium bomb will leave different radioactive byproducts than a plutonium bomb.
To break it down to layman terms, nuclear explosions are also designed to emit energy extremely fast, meaning the radioactive chain reaction "burns" through elements very fast. This is different that the fuel in a nuclear reactor, which is designed to burn hot and slow, meaning there are more longer-lasting byproducts left over and why Chernobyl is a no go zone for thousands of years, but we can live in Hiroshima.
Most of the cancers that happened were from people downwind from the explosion. Most of these elements that caused the cancers and sicknesses decayed away within a couple of years.
> 2. Would exploding the equivalent amount of TNT look exactly the same?
No. It wouldn't produce anywhere near the amount of heat/light. The TNT equivalent is usually used to measure the destructive force equivalent of the explosion.
> 3. Would USA fake having a single bomb that destroys an entire city?
The US may "fake" having a number operating bombs ready, etc. But obviously there's no need to fake it as the US destroyed 2 cities at the end of world war 2 and exploded hundreds of test bombs since.
> 4. What happened to the deadly radioactivity in Japan?
The bombs dropped were exploded high in the atmosphere to spread the explosive force of the bombs. Most of the radioactive material was carried away by the winds and/or had a short half-life. Most radioactive material from the bombs decayed away and there is no longer a statistically significant higher risk of cancer in either Hiroshima or Nagasaki.
> 5. What is carpet bombing?
It's when you have a fleet of bombers drop massive amounts of traditional bombs on a city, as was done to Germany and Japan during world war 2 (and by the Germans to a few cities like Rotterdam).
> 6. Would USA fake having a single bomb that destroys an entire city?
Same as question 3.