1) '별보다 빛나고 아름다운 당신의 미소 때문에 ; 별보다 빛나고 아름다운 세상을/사랑을 알았어요'
So, 'Because of your beautiful smile that shines brighter than the stars ; I understood a world/love that shines brighter than the stars'?
I'm not sure how to translate '알았어요' in this context.. '^'
-> '알았어요' is from the verb '알다.' so you can translate it into I got to know/understand a world~~
2) The other is : '다 나아질 거라고' vs '모두 좋아질 거라고'
Both are used in the song. Do they have the same meaning or is one more like 'Everything will get better' and the other 'Everything will be okay'?
=> 모두/다 has the same meaning (like everything/all). 나아질/좋아질 also has the same meaning. (original form is 나아지다/좋아지다.) so those two don't have the meaning difference. you can translate it into the same sentence.
In case of 'everyting will be okay', it could be expressed as '모두 괜찮아질 거라고'