Sure, here are the corrections for the underlined parts where necessary:
Did you hear about Sophie? She has given up her job. (Present perfect is appropriate because the action affects the present, but "is" could be used if referring to a recent change.)
My mother grew up in Italy. (The correct tense is simple past, as "grow up" refers to a completed action in the past.)
How many plays has William Shakespeare written? (This sentence is correct as it uses the present perfect.)
I've forgotten his name. Is it Joe or Jack? (This sentence is correct.)
Who invented paper? (Use the simple past because it refers to a completed action in history.)
Drugs are a big problem everywhere. (Present simple is appropriate here.)
We've washed the car, but now it's dirty again. (This sentence is correct.)
Where were you born? (The simple past is appropriate for a completed action in the past.)
Ellie has gone shopping. She'll be back in about an hour. (This sentence is correct.)
Albert Einstein was the scientist who developed the theory of relativity. (Use the simple past for both verbs because they refer to completed actions.)
Let me know if you have any further questions!
