1 What is inspiring him? 2 Why does he always feel lonely? 3 Why does he admire Alexander Pushkin?

Your tenses are very confused. Are we talking about past or present time? 1. What inspires him? That's fine, if you mean this to be in the present tense. However, if we are talking about the past, then it should be: What inspired him? 2. Why is he always felt lonely? This is ungrammatical and unclear. How you correct it would depend on the meaning intended. It could be: Why does he always feel lonely? Why has he always felt lonely? Why did he always feel lonely? We would need to know whether we are talking about the past, the present, or both. 3. Why did he admired Alexander Pushkin? This is also ungrammatical. If you want to say this in the past, it would be: Why did he admire Alexander Pushkin?