1. What's he bought?
2. What has he bought?
[Are both okay?]
3. How's he solve the problem?
4. How has he solve the problem?
[Are both okay?]
5. Why's he got angry?
6. Why has he got angry?
[Are both okay? Thank you for your help. Have a good time.]
1 is OK in casual speech. 3 and 5 are incorrect.
2 is correct
4 is incorrect. It should be "How has he solved the problem?"
6 is incorrect. It should be "Qhy did he get angry?" or "Why is he angry?"