I only know of one firm that lets a significant portion of its junior associates work in its offices abroad. Many firms with international offices hire European attorneys, although they may let you split your summer abroad and trick you into thinking you may be able work abroad. You can go to a firm's website and look at the bios of the attorneys in international offices and see whether they have US JDs. You could always do JAG if you want to be abroad.

What exactly is international business law? Representing international companies? You do that when you're a corporate associate at any major firm and it's not called international business law. Is it working abroad for companies? Legal fields that affect companies that transact business abroad, like international trade law?

Background is totally irrelevant. None of the securities lawyers I know at the biglaw firm I work at have business degrees. None of them even took securities in law school. You learn what you need to know as an associate.

I don't think taking a bunch of bar classes makes any difference, you're certainly not going to remember the black letter from 2 years ago when you're studying for the bar and you don't need to, everyone takes barbri. Take some or all of the basics - evidence, corporations, criminal law/procedure, tax, constitutional law II, and then take things you are interested in. You can learn family law when you're studying for barbri, but you'll likely never have the opportunity to learn Islamic law, literature and the law, film first amendment law, law and national security, history of jurisprudence, animal law, etc again. 

I know plenty of people who didn't take more than 1 or 2 bar classes and got biglaw jobs, clerkships, and passed the bar.