겠다 vs 나 보다

1. I see a little girl crying on the streets: 길을 잃었나 봐

2. My brother calls me (and he only ever calls if he needs something): 문제가 있나 봐

3. My cousin fell off his bike and is crying: 아프나 봐  (*)

4. There’s an ambulance rushing: 사고가 났나 봐

5. I saw a mailman from the window: 소포가 도착했나 봐

6. There’s a cool poster of a new movie: 재미있겠네

7. Someone fell from the stairs (i don’t know if she is crying or not): 아프겠다

8. Someone is competing in a spelling bee: 떨리겠다

9. Someone got a prize: 행복하겠다 (Okay, but 좋겠다 is more natural)

10. I go to a thrift store and see a vintage Chanel bag: 비싸겠다

11. Someone is competing in a math contest: 똑똑하나 봐  (Slightly unnatural because it is too obvious)

12. My friend tells me she has a girlfriend: 여자친구가 예쁘겠다

-나 봐 and -ㄴ가 봐 indicate inferring or guessing (involves deeper emotional or thinking process), while -겠다 is more a natural, spontaneous reaction with very little thinking.

So seeing someone fall off a bike is a good example.  If you just saw it and the person is getting up immediately, you'd probably say 아프겠다 as a casual reaction.  But if the person is crying or looks hurt, you'd say 아프나봐/아픈가 봐, because now you're sympathizing with them - in this case, the phrase is more like (굉장히/너무) 아프나 봐, like you're worried.

Anyway, most of what you have is correct and natural.

겠다 vs 나 보다

1. I see a little girl crying on the streets: 길을 잃었나 봐

2. My brother calls me (and he only ever calls if he needs something): 문제가 있나 봐

3. My cousin fell off his bike and is crying: 아프나 봐 아픈가 봐

4. There’s an ambulance rushing: 사고가 났나 봐

5. I saw a mailman from the window: 소포가 도착했나 봐

6. There’s a cool poster of a new movie: 재미있겠네

7. Someone fell from the stairs (i don’t know if she is crying or not): 아프겠다

8. Someone is competing in a spelling bee: 떨리겠다

9. Someone got a prize: 행복하겠다

10. I go to a thrift store and see a vintage Chanel bag: 비싸겠다

11. Someone is competing in a math contest: 똑똑하나 봐

12. My friend tells me she has a girlfriend: 여자친구가 예쁘겠다