And I think the answer to both questions is "no". Williams was convicted based on the testimony of self-interested informants. It is routine for such testimony to be false. It is routine for jailed informants or informants accused of a crime to routinely make up confessions of such crimes. One famous case was a defendent placed in a cell with someone he knew to be a "snitch". He told his defense attorney he had not intention of saying anything, and (I believe at his defense attorney's suggestion) placed tape over his mouth to avoid false stories. The next morning the informant told a story of the defendant having confessed in writing, and then flushed the confession down the toilet or eaten it or destroyed it in some other way.
No explicit deal has to be made between a prosecutor and such informants, though often they are. All a prosecutor has to do is put the word out that he is after such and such a person, and it becomes known that plausible information against them (true or false) will be rewarded. Releasing copies of the defendants file to likely sources is a common way of obtaining such plausability. One of the reasons "snitches" are so hated is that there is a whole culture of informing out there - often based on lies and injustice. (Aside from actual deception, informing leads to injustice because often the most guilty and most hardened criminal will be the one to inform against people he or she persuaded to take part in the crime; so the worst criminal ends up with the lightest sentence.) Many prosecutors don't even think about "truth" and 'falsity" but "good case" and "bad case". Also the there is the attitude that the guy they are going after is a bad guy; the crime charged will do to put them away.
Obviously this is a problem in criminal cases in general , not just death penalty cases. It used to a tradition in pre-Christian Rome that Emperors who wanted to be popular and gain a reputation as reformers would proclaim an end to the use of informers. But paid informers were too critical to Rome's system of governance; if a serious attempt was ever made to do without them it did not last long.
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