1.The issue about whether the students should be taught to cooperate or compete is of great interest to many people.
The issue, whether students should be taught to cooperate or compete, is of great interest to many people.
Punctuation clarifies the sentence.
2.The discussion of which is more important to the people, the government or the others' donation is of concern for many people.
The discussion about the relative importance of government funding and charitable support is important to many people.
Your sentence is mostly grammatically correct except you don't need an article or a possessive for the others', and donation should be plural. You don't need 'to the people'. Your closing, 'for many people' makes it redundant. You could reduce the sentence further by saying, 'public or private funding', instead of 'the government or the others' donation'.