the intent behind those words was not given

"The LORD set a mark upon Cain, lest any finding him should kill him". Again, I don't see how God could have possibly made it any clearer that the intent of putting the mark on Cain was to prevent the otherwise very real possibility of people killing him.

I think it's "kill them and six members of their clan/family", but I'm not sure.

If you're not sure, then you must believe that there could be circumstances under which killing six members of a person's family as punishment for a crime they did not commit could be justified. I find that deeply disturbing.

the first part seems, to me, to refer to wicked deeds

No, it simply refers to an evil state of being. It says nothing about what brought about that state. But it doesn't matter. The fact that it specifically calls out thoughts means that the Flood was at least partially retribution for thought crimes.

But my moral intuitions are also, to a large degree, a product of my environment, and specifically of my upbringing.

Sure, and so are everyone else's.

my moral intuition is closer to God's Word than it would have been had I been raised in a different culture

A Muslim would disagree with you. Have you considered the possibility that they might be right and you are wrong? It's just the luck of the draw that you happened to be born into a Christian household rather than a Muslim one. Maybe you got unlucky. How would you tell?

But you keep dancing around the real question: Do you really believe that killing innocent bystanders can be morally justified? Or that genocide as a response to thought crimes can be morally justified? Or that forcing people to cannibalize their own children (Jeremiah 19:9) can be morally justified? Because that is the price of taking the Bible as your moral standard.