But must I spend it instantly or can I take it away, invest it elsewhere and those earnings also be tax-free?
It's worth pointing out that contributions can be withdrawn from a Roth anytime with no penalties. Earnings can be withdrawn penalty-free under some conditions, like a first-time home purchase.
You don't have to spend it instantly, but earnings on investing it elsewhere (not in a tax-deferred account) would be treated like any other taxable earnings.
Are earnings tax-free only if they occur while inside the Roth? Or, having passed through the Roth, are they always off the radar?
Only in the Roth. They're not taxed when they're withdrawn from the Roth (if done according to the rules), but any subsequent earnings on those former Roth funds (outside the Roth) could be taxed like any other investment.
Does the Roth have to stay in the US?
Good one! Not a clue. As you've probably learned already, many U.S. fund companies have restrictions on what addresses they want to deal with. As custodians of a Roth, they may also limit the investments by type to be more restrictive than the tax code. For example the IRS forbids many types of self-dealing in IRAs so the custodian does too. But although the IRS doesn't seem to have a position on minors holding Roth IRAs, Fidelity won't be a custodian of a minor's Roth IRA yet T. Rowe Price will.
I bet most custodians will ask that your Roth have a U.S. address and hold U.S. investments (or through ADRs, ETFs, or mutual funds).
Can I remove money from the Roth and take it offshore where it can also grow tax-free?
Sure. Contributions anytime, earnings penalty-free under various restrictive conditions. But the money won't be in a Roth account any more, and it might be subject to some sort of tax reciprocity.
Have you read any of Ed Slott's books? I don't know if he specifically addresses the concept of non-U.S. investments in Roth IRAs, but his
discussion board has probably seen that before. A couple of CPAs post there who will tackle any question... especially Denise Appleby & Bruce Steiner.