Should the nature of the crime affect a defendants opportunity to plea bargain?A plea bargain is another controversial practice in the American...

In reality the nature of the crime does indeed sometimes determine whether or not a criminal defendant can plea bargain. A plea bargain is an agreement between the prosecutor(the State) and the defendant. There are no hard and fast rules regarding plea bargaining. One side, either the defendant or the district attorney, says look... let's make a deal...the DA may say something like if you(defendant) will plead guilty to XY...then the State(prosecutor) will do something(like only charge you with XX). Or the defendant, and his counsel, will say... look if I plead guilty to BC...then you will agree not to charge me with GP. Either party can try to initiate a plea bargain(deal), neither party has to accept it. Both parties must agree to the terms of the deal(plea bargain). If they can't agree, then no plea bargain is done and the case will proceed to trial. This is never one sided, in other words, if the defendant says... I want a plea bargain...the State(DA) can say no...we are going to trial and prosecute you. If the State(DA) says...let's plea bargain...the defendant can say no...I want to go to trial because I didn't do the crime or I think I can win the case. If the State thinks they can win, they will not offer a PB. If the defendant thinks he can win he will not want to PB. 

Now, if the defendant was in fact wrongly accused, as you put it, and found guilty, but then later new evidence surfaced to exonerate him, then his lawyer must petition the court to hear this new evidence. If that request is granted and if it does exonerate him, then we can get him out of jail.

You mentioned that the sentence was the death penalty. All capital cases automatically come up for appellate review.