Is it possible for me to, for example, purchase address space from APNIC and then use those addresses in Europe or the United States? Would there be a cost or paperwork involved?
From the APNIC FAQ:
Basically, it's possible, but there's a risk some users won't be able to access the resources if they're located worldwide.
There's also the problem of requirements for getting the IPs from APNIC in the first place but that's a different story.
The short answer is yes. Which registry you get your resources from is supposed to be based on where you are located, not where your equipment is located.