1. They didn't believe before, but they believed after that because they asked 'so what can we do now' of Jor El... So, they knew it was going to explode.. just not exactly when..
2. If it was indeed same day (not saying that it is), it would really be another plot hole.. the coincidences are just too much... Jor El plans to have his baby shipped off, and it was born on the day the planet explodes.. hahahahahhahahhahah... and then the revolution by Zod also happens on that exact same day... and the council sends them to the phantom zone on that day... holy molly...

(note: goyer needs to enroll himself into a 'critical thinking' class)
3. Did they show the DP falling? Not sure about that even though I have seen it 3 times... but think about this, China is building the world's tallest building 220 stories, that can support a small city (30,000 people), in 90 days... so, if they employed the Chinese do do cleanup, the entire Metropolis could be rebuilt in 1 year..