Such is also a ticklish sentence for me to "eat":
A charge of throwing a bomb with intent to kill, which charge carries the death penalty, was dropped by the Crown. (original)
Question:
Is the original grammatically correct?
Do both charge's have the same meaning? (I don't think so but I…).

The "Crown" is a name of a person, but why is the "the" put before "Crown"?
My guesses are:
It is grammatically correct. The first charge means a task or somewhat a violent attack and the second charge more a crime than a violent attack. I would rewrite the original this way:
The Crown dropped a violent attack of throwing a bomb with intent to kill, this （ or this action, that action of his) carries the death penalty.
Is my guess correct?

Question

Such is also a ticklish sentence for me to "eat":
A charge of throwing a bomb with intent to kill, which charge carries the death penalty, was dropped by the Crown. (original)
Question:
Is the original grammatically correct? 
Do both charge's have the same meaning?  (I don't think so but I…).

The "Crown" is a name of a person, but why is the "the" put before "Crown"?
My guesses are:
 It is  grammatically correct. The first charge means a task or somewhat a violent attack and the second charge more a crime than a violent attack. I would rewrite the original this way:
 The Crown dropped a violent attack of throwing a bomb with intent to kill, this  （ or this action, that action of his) carries the death penalty.
Is my guess correct?

Answer

It would be better to omit the second "charge" but the sentence is correct as it is.

I would say that the two uses of "charge" have the same meaning.  A "charge" is an accusation.  Once charged, a defendant can be found to be either innocent or guilty.  If guilty, he pays the penalty.  When we say "A charge of jaywalking carries a penalty of a five dollar fine", it is implicitly understood that the fine will only be paid if the defendant is guilty of the crime.

Answer

1. Yes, the original is grammatically correct.
2. No, the two "charge"s do not have the same meaning. The first refers to an accusation, the second to the specific crime.
3. "The Crown" refers to the state or government, not a person, hence "the" is used.
4. Your rewrite changes the meaning. "Charge" in this context means legal accusation, not an attack.