How could I think that God could approve of the death penalty?

Well, let me see, on the cross, I don't recall Jesus rebuking anyone for putting to death the two criminals on either side of him.  If I recall correctly, one of the criminals rebukes the other calling their punishment "just."  

As for "Let he who is without sin, cast the first stone" that is the most misunderstand passage in the Bible. In order to execute someone, the law of Moses called for two or three witnesses.  There many have only been one--which means they cannot stone her.  Second, even if they had the required witnesses, the law called for the stoning of both the woman and the man.  The man was not there.  The entire situation was trumped up.  The law of Moses actually demanded that they let the woman go! 

Finally, they were in no position to condemn her, since many of the people wanting to stone her were in ongoing adulterous relationships.  Hence "Let he who is without sin..."