Does this course have no listed textbook or is the professor teaching entirely from notes? That would really help pin down the level.
If the exam problems are as you describe, I think all those GR books (Carroll, Hartle, Schutz) might be too advanced and more relevant would be something like Taylor & Wheeler: https://www.amazon.com/Exploring-Black-Holes-Introduction-Relativity/dp/020138423X
Agreed. Griffiths has the best books for undergrad level E&M, Quantum, and particle physics. I cant recommend any of the classical mechanics or thermal books. if you want to try General relativity you can start with Taylor and Wheeler: http://www.amazon.com/Exploring-Black-Holes-Introduction-Relativity/dp/020138423X
Find it in a library, its a bit small for that price, but I recommend:
https://www.amazon.com/Exploring-Black-Holes-Introduction-Relativity/dp/020138423X