"How does he/she look?" vs "What does he/she look like?"

Third option: how is she looking [these days]? Used to ask for an update on someone you haven't seen in awhile and whose condition you expect to be in flux.

How do they look means you want to know overall condition, usually on a scale of comparison. Emotionally, how do they seem to be feeling? Physically, do they look okay? Is today better than yesterday? What do they look like is an inquiry into physical features - hair color and length and texture and style, eye color and shape, nose shape, skin color, etc.

Yeah, your usage is correct

It depends on the context. The first one can also be used for physical appearance (usually when they put on new clothes cosmetics).  Like if my wife dressed my son in a suit he hasn't worn in a while, she might say, "how does he look (in the suit)?" 

"What does she look like?" is asking for a factual description of her. What color is her hair? Does she have clear skin? Is she tall or short? How much does she weigh, roughly? What was she wearing last time you saw her? These are all facts that might help someone recognize her if they saw her. "How does she look?" is asking you to make a judgment call about her appearance. You might find answers to this question that provide factual information, but you'll usually find a theme among the observations that tells you what judgment the person is making. For example, "she's beautiful," is a judgement; meanwhile, "she's got three broken ribs, a black eye, two missing teeth, and a cut straight through her lip," *sounds* factual, but all observations are made in support of a specific judgment: "she's really badly hurt." So, the answer is still ultimately a judgment.

I would say 'how does he/she/it look' is more general, it can be referring to both physical and emotional traits. It's a bit like asking 'what impression do you get when you look at them'. 'What does he look like' is more literal; height, weight, hair colour, clothes etc.

I’d say how does “x” look/feel/taste/smell/sound is asking for either the quality (good/ok/bad) or a description of its current state. Your example of someone looking sick works for the second option, so for the first option you could say for example “I just cleaned my house, how does it look?”. What does “x” look/feel/taste/smell/sound like is asking for a description of more permanent traits. “What does he look like?” He’s tall, handsome, skinny, has long hair, etc. things that don’t change or at least are more difficult to change (the hair thing is a bad example of that but it still falls into this category)

Your usage is correct. I use them that way also.

Yes, your usage is correct. But to answer your second question, also yes, there are other ways to use those sentences. "How does he/she look?" in particular can mean both "What is his/her appearance like" and "By what means does he/she look at things". There's an old joke that takes advantage of this ambiguity: "I know a man with no nose." "Really? How does he smell?" "Terrible."

"How does he look" isn't a common phrase at all. It's fine, but quite difficult to think of an appropriate context. The most common usage of "how...look" is asking about yourself - asking someone if your choice of clothing is appropriate to the occasion, "How do I look?" We also commonly say, "Do I look OK?" - seeking their opinion about e.g. whether my skirt is too short, or I've screwed up my mascara. I suppose I could say "how does he look" if I'm on the phone to my wife, asking about a costume that our son is wearing to a party. It's quite a niche expression. "What does he look like?" is usually asking for a physical description - if he's tall, fat, bald, etc. for identification purposes. It's a common question.

"How does he/she look?" = Asking for an opinion of their appearance. Do you think they look smartly dressed? Cute perhaps? "What does he/she look like?" = Asking for a description of someone. Do they have black hair? Brown hair? Are they tall? Are they thin? Anything that will help form an image in someone's mind.

You've got the jist, yeah. To sum up "What does he look like" is asking for an objective, factual description of him. "He's 6' tall with brown hair and green eyes." "How does he look" is asking for a subjective judgement of his appearance. "He looks a little sick." Or "He looks great!". It's not necessarily just does he look sick or healthy. The judgement being asked for is context dependent. Like someone could be asking about his outfit or general attractiveness. "How does he look?" "He looks pretty hot." Etc.

“How does s/he look?” Is asking for some qualitative assessment of the person’s appearance or condition. This could be asking whether they look good or bad; or, if asked to a medical professional, could be asking about the prognosis of a patient. “What does s/he look like?” Is asking for a physical description of them.

Use "what" with like, not "how". We don't use "how" to ask the description of something. How is the weather?...What is the weather like? How is London?...What is London like?