Here are the corrections for the underlined parts of the sentences where necessary:

3. How many plays **has William Shakespeare written**?
- **OK**

4. **I’ve forgotten** his name. Is it Joe or Jack?
- **OK**

5. Who **has invented** paper?
- Correction: "Who **invented** paper?" (The present perfect tense is not appropriate here because paper was invented in the past, and there's no relevance to the present.)

6. **Drugs have become** a big problem everywhere.
- **OK**

7. **We’ve washed** the car, but now it’s dirty again.
- **OK**

8. Where **have you been born**?
- Correction: "Where **were you born**?" (Again, the past simple tense is more appropriate as it refers to a specific event in the past.)

9. **Ellie has gone** shopping. She’ll be back in about an hour.
- **OK**

10. **Albert Einstein has been** the scientist who **has developed** the theory of relativity.
- Correction for first part: "Albert Einstein **was** the scientist who has developed the theory of relativity." (The first part uses past simple because it refers to Einstein's existence and accomplishment in the past. The second part can remain in present perfect as it describes an ongoing relevance of the development.) Alternatively you might say "Albert Einstein **is** the scientist who **developed** the theory of relativity."

Revised sentences:
3. How many plays has William Shakespeare written?
4. I’ve forgotten his name. Is it Joe or Jack?
5. Who invented paper?
6. Drugs have become a big problem everywhere.
7. We’ve washed the car, but now it’s dirty again.
8. Where were you born?
9. Ellie has gone shopping. She’ll be back in about an hour.
10. Albert Einstein was the scientist who has developed the theory of relativity. (or "is the scientist who developed...")