Yes, they are punishable for violating any of these seven mitzvos; none of them are more or less severe than the others. The punishment for each of them is decapitation (Rambam, Hilchos Melachim 9:14).
No warning is needed, but there needs to be at least one male witness and at least one male judge (ibid.). The judge can be either a Jew or a non-Jew (ibid. 10:11).
If he committed the violation in error, he's not punished. "Error" here means that he was unaware that the object is a forbidden one (for example, having relations with a woman under the impression that she's unmarried or is his own wife, when in fact she's married to someone else), but ignorance of the law is no excuse (ibid. 10:1).