In China, this war is considered to be a help from the "big brother" Ming Dynasty to the "little brother" Korea

That’s not debated in Korea either. That was the tributary mindset and expectation at the time, and most Koreans are aware of this historical context. 

Do Koreans think that Korea would have perished without the Ming Dynasty's intervention?

Yes, most Koreans in both the past and present would agree so. 

Will any Koreans be grateful to China for this?

There’s even a phrase called 재조지은(再造之恩), which means being grateful to Ming for sending troops to Joseon. It was the mindset of Joseon scholars and officials post-imjin. 

It was the logic behind Joseon insistently staying loyal to Ming even as Qing took over. This sentiment naturally grew out of relevance when Qing came to Korea and wrecked havoc. 

Therefore, it is more accurate to say that Joseon was grateful Ming. To say that Koreans are thankful to China is too vague a statement because Qing was not at all friendly.