Not leaving the country would probably be reasonable, while leaving the country maybe much more difficult (relating to a security clearance). Talk to your recruiter. As with BDCP the main issue at hand is keeping your graduation date with the degree in your contract.
Doing an internship abroad could bring up a lot of problems like preventing you from getting a required security clearance for your designator, maintaining the required contact with your recruiter, taking the PRT every 6 months and overall accountability for you.
As for your question about getting paid, you would get paid. You enlist the Naval Reserve while in BDCP and then become entitled to basic pay, BAH and BAS.
Overall, talk to your recruiter. Keep in mind for planning purposes that you can change you academic degree completion program easily before you apply, but once you are in it become very difficult.