The President has exclusive power to negotiate with foreign countries, per Article II, Section 2, Clause 2 of the Constitution. He has exclusive power to appoint ambassadors, permanent or otherwise. He has control over visas. He thus has power to approve or deny congressional visits to a hostile nuclear power, especially in time of war. You will find no enumerated powers in the Constitution authorizing members of Congress to negotiate with a foreign government.
Exactly right. She can't negotiate with the North Koreans, and they know it. They might be trying to cast her in the role of "useful idiot" for publicity purposes, but that's riskier for her than it is for the Admin.
Historically, the President has not tried to stop Congress from taking foreign junkets, even to countries where most citizens are barred from visiting. With everything on his plate, it would be silly for Pres. Bush to spend any of his political capital on such a meaningless fight.