I want the trustee to be based in the USA (although everything else is in Australia). Is this allowed or does it have to be an Australian trustee?
It's "allowed", but it means that the trust will be a "non-resident trust" for tax purposes (and this can lead to bad, or at least inconvenient, tax consequences).
Generally when we tell people this, they then change the trustee.
Refer to the following page on the Tax Office's website for more information:
Residency requirements for trusts: