Answers:
1. She doesn’t look very well today.
"She" is a singular pronoun, so you should use a singular verb.
2. Interesting news is what sells our paper.
"news" is a singular noun, one of the exceptions which ends with s.
3. They don't think they’ll win the game tonight.
"They" is a plural pronoun.
4. It doesn't seem so cold today.
"It" is a singular pronoun.
5. A pound of cookies costs about a dollar.
in this case, the subject of the sentence is not the cookies (plural) but rather the pound (singular). thus, the verb should be singular.
6. Gymnastics is easy for Angela.
like in question 2, "gymnastics" is a singular noun. don't let the s at the end fool you!
7. He doesn't speak very well.
"He" is a singular pronoun.
8. Twenty minutes is the amount of time it takes me to get home from work.
this is another tricky one! "Twenty minutes" is actually referring to one block of 20 minutes, or one period of time, thus it is singular.
9. One hundred dollars is not a lot of money to some people.
like the question above, "one hundred dollars" is a collective noun and is singular.
10. Mumps is one of the most uncomfortable diseases.
the explanation for this is similar to q2 and q6.
i hope this helps! :D