1. She had three sons, and they were happy family.

(Is this a right sentence?)

2. One day she got a disease and stayed in bed. (Is it OK?)

3. What will you do 20 years later?
3-1. I will teach math at an academy.
3-2. I will be studying abroad.
3-3. I will start to do business.
3-4. I will be able to buy a new car.
(Are the answers all correct?)

3 answers

All sound fine.
Everything looks correct. However, I would change #2 to say "One day she because ill and stayed in bed."
I think GuruBlue meant to write, "One day she became ill ... "