Generally speaking, you must obey the law of the country you’re in and in some cases you must also obey the law where the servers are located.
It is often a good idea to think about the law of the country which will be most impacted by what you were doing (if you’re creating a website that will be heavily used by citizens of another country, for example), and try to follow relevant international laws.
There is often confusion in the US about fair dealing (in the UK and most Commonwealth countries) and moral rights (in the EU).
Fair dealing is not the same as fair use. There are no fair use provisions in most international law. Fair dealing tends to be much more specific about how you can make educational use of material.
Moral rights are generally not recognized in US (except in very specific circumstances), though they part of the Berne convention and also widely part of the law in many countries.
In some countries, particularly in the EU, databases can also be protected. So, when using or creating databases for use in other jurisdictions beyond the US, there may be some additional copyright protections to be aware of.