"How is he?"
他=He 怎么样=How 他怎么样?= He (is) how? Because, remember, we don't need the character 是 when were asking about an adjective. And when you form question sentences in Chinese, you're basically putting them in a statement structure, so you say "He is how?" When you're asking somebody, "你好吗?" you're not asking them about their physical wellness. Some people just say 你好 as a greeting, literally telling them that they are good. When you ask, "你好吗?" you're literally asking if the person is good. It's not an in-depth look into their life, it's not asking about their physical ailments. It's just asking if they're good. When you ask, "他怎么样," you're asking, "How is he?" As in, "How's his health/situation?" It's sort of denotes that you're asking for a more in-depth answer.
他好吗？and 他很怎么样？are both ideomatic, the first meaning "Is he (doing) well/fine/OK." and the second one "How is he?"
No, 他很怎么样？should not be accepted. It is not idiomatic, it translates to "He is very how?" and does not mean "How is he?"
But In a situation, where someone says 他很浪漫。and you did not understand the last word, you could say 他很 ... 什么? - "He is (very) ... what?"
他/她好吗？and 你好吗？are not commonly used by Chinese. Western people are used to ask "how are you?" but not them. If you ask them 你好吗？They will understand but feel weird about it, they only ask that if they know you are sick or something. Instead of that use 你吃饭了吗？it's weird for us but is the Chinese way to ask how are you.
As a Chinese I must make some comments. It's true we don't always say 你好吗？but it's not weird at all when used at the right time, say seeing someone who you would call a friend but not a close one after not seeing him for a while. OTOH it would be likely to sound awkward to ask someone you meet for the first time if he has eaten rice yet. Especially not at a business lunch, LOL.