Our sages teach us that a person is responsible for his actions. Consequently, he is obligated to pay for any damage that he personally perpetrates, irrespective of whether he did it purposely, or inadvertently.

1 + 2) For damaging a limb of a slave, either way, he must therefore pay the master for the depreciation, as per the Beis Din's assessment.

You used the term 'by accident.' Notice, that I avoided using it. If the damage is caused by a pure accident (for example, if Reuven lies down beside Shimon who is already asleep, and Shimon stretches and strikes him, breaking his nose in the process, he is Patur (absolved) from paying.

3) What I wrote does not extend to the death sentence, which one only receives for an act that is performed on purpose. Generally, someone who commits a sin that carries the death sentence is obligated to bring a sin offering. Murder is an exception, as the Torah obligates a murderer to go into exile to one of the cities of refuge. Depending on a number of factors, he is sometimes absolved from this punishment. Sometimes, he remains at the mercy of the closest relative, who is permitted to kill him. In any case, in most cases he is not exempt from punishment, as you suggest.

Be'Virchas Kol Tuv

Eliezer Chrysler