-게 되다 means one has come to a point to embark on something good or to be subjected to something bad. Your sentence is using it very naturally. Alternatively, you could also say: 드디어 외국으로 유학(을) 가게 됐어요, where 유학(을) 가다 is a common phrase for going far away from home for one's study (유학하다 is a more formal version of it). So going to a foreign country to study is typically said as "X(country)-에/으로 유학(을) 가다 [or 유학하다 more formally]". But 외국에서/X-에서 공부하다 is also very common.

Yes this is correct. Ex. I finally get to eat dinner. 드디어 저녁 식사를 하게 됐어요.

looks good! you could alternatively use 유학하다 which means to study abroad 드디어 유학하게 됐어요!