A lot of joint programs I've seen require the LSAT or the GRE. You still have to apply to each program separately, but it seems as if admissions will accept the LSAT in lieu of the GRE if you are applying for a dual program.
Personally found my Biology background to be an advantage on the LSAT, particularly the reading comp. section for all the science passages. However, I agree with other posters that if you are going for a dual degree then take the GRE.
Also worth checking out study abroad/dual degrees at schools that have a strong international focus like at NYU and Columbia. For example, Columbia has FANTASTIC options to get LLMs in other countries.
I think the biggest argument for the GRE is for dual degrees (like JD/MBA). I think it makes sense that law schools would accept it in that scenario, allowing applicants to only take a single exam. But otherwise, it makes more sense to take the LSAT.
I was in the same position but no you don't need to worry about anything like this. As long as you have dual citizenship, you'll pay the "local" rate but obviously out of state residence fees. You wouldn't be subject to international fees.