Whilst much of the science in the film is flat wrong and even nonsensical, the idea that bombs could send Earth spiralling into the sun does have some basis in reality.
For an object to orbit something there has to be a gravitational force between them, and the orbiting object has to have some sideways speed. That's what you see happening when a rocket launches - watch any launch and the rocket tilts over to the side almost straight away, and keeps tilting over as it goes up. This is because the rocket needs to both climb out of the atmosphere and gain sideways speed in order to go into orbit.
The radius of the orbit is determined by how fast you are going towards the side. Basically the slower you go the closer you orbit.
What the bombs did, then, was slow the Earth. This would have the effect of reducing the orbital radius. What's important is that it would not just keep getting closer and closer until it fell into the sun... rather it would just shift to take up a new stable orbit that was closer to the sun. And yes, that would happen over a period of months.
The corrective bombs would do the same in reverse. They wouldn't make the Earth go speeding ever outwards forever, but rather speed up the Earth to its original speed and so shift it back into its original orbit.
Of course in reality no bomb could really change the Earth's orbital speed. But if it could, then the change in orbit isn't actually that far off reality.
You Are Wrong. The Closer a Satellite is to the Body it is orbiting, the faster it much go to remain in obit.
The is most event in the space shuttle. Obriting about 200 miles above the Earth it goes at 17,500 miles per hour...but MOON is 250,000 miles away and cirles the Earth at about 2,000 miles an hour and take 28 days to complete on orbit.