2 and 3 might be heard but not 1. "Why noun?" is OK. "Why subject verb?" isn't.

Yes, these are natural in casual conversation but not grammatically correct. "Why suddenly he left?" → Informal. Correct: "Why did he suddenly leave?" "Why suddenly a trip to India?" → Natural in speech. Correct: "Why a sudden trip to India?" "Why suddenly donuts?" → Very natural informally. Correct: "Why are you talking about donuts suddenly?" Informal speech often skips words for brevity.

I'll omit "suddenly" and other words that have no effect on the discussion. Your phrases are then 1. Why he left? 2. Why a trip? 3. Why donuts? None is a sentence, but #2 and #3 are useful as sentence fragments. Without the question mark, "why he left" can be useful as part of a sentence: "I don't know why he left" "Why he left is a mystery to me" "Why he left, I know not." It can be part of a question, but extra words are needed to create the question: "Do you know why he left?" #1 can also be rescued by breaking it into two sentences: "He left. Why?" Yes, "Why?" is a complete sentence.

For 1, it would be more natural to say "Why did he suddenly leave?". For 3, I would say "Why donuts all of a sudden?".