Actually the earthquake was predicted
If by predicted an earthquake, you mean he predicted the wrong time and the wrong place, then yes, he predicted an earthquake. And this is not the first time he has predicted an earthquake, this is just the first time his prediction was within a week of an earthquake actually occurring.
Giuliani uses radon as a measure of earths movement, and tries to use increased radon levels as a sign of an impending earthquake. This method has never been found to predict earthquakes, but even a broken clock is correct twice a day.
How is it these guys can be drilling again?
So IODP expedition 308 took place in 2005, some time before the GoM spill. Did you even read the site you linked?
And "these guys" are an NSF funded research organization, not an oil company.
I sailed on IODP expedition 308 (and 304, the expedition the article actually refers to) and one of the objectives was to learn how to safely drill in overpressurized environments safely. And the best place to do this is somewhere where you have ample geophysical data relating to sediment formations to identify and avoid potentially dangerous gas and oil bearing sediment. It turns out the organizations that have that geophysical data is the oil companies. If the USGS or DOE or PETA had that information, they would have collaborated with them, but they don't so we didn't. The sites drilled during EXP 308 were specifically chosen to avoid such dangerous places. Every sample that was brought aboard was measured for higher order hydrocarbons which are indicative of thermogenic gas and oil. Once those values reach a certain (very conservative) threshold drilling stops, and the hole is filled with heavy mud. However I don't believe we ever came close to that threshold.
The point being that researcher-industry collaborations are not inherently bad because industry is involved. I thought it was a good thing that industry was be interested in making drilling in the GoM safer.
The carbon would come from the atmosphere and go back.
How exactly does atmospheric carbon penetrate the kilometers of sediment and rock needed to reach most oceanic gabbros?
I have a theory that it's impossible to prove anything, but I can't prove it.