Here are the corrections needed for each sentence:

1. **My sister wondered where bought the best cakes. (to buy)**
- Correction: **B** (should be "where **to buy**")

2. **She might eat too much before the race if she wants to perform well. (might not)**
- Correction: **A** (should be "might **not**")

3. **What was she do while her mother was making lunch? (did)**
- Correction: **B** (should be "was she **doing**")

4. **The bigger the city is, the more noisy it is. (nosier)**
- Correction: **C** (should be "the more **nosier**" or "noisier")

5. **We ran down of milk, so I need to buy some more. (ran of milk)**
- Correction: **C** (should be "ran **out** of milk")

In summary:
1. B
2. A
3. B
4. C
5. C