Себя is reflexive, meaning more like "self" than "you." It indicates the subject of the sentence. A person can use it to refer to themself, as well. For example, if a man said "I love my wife" he could use a form of "себя" in place of "my." ("Я люблю свою жену.")
Literally, the sentence is, "how (do) you self feel?"
I will try to relay an answer that i've seen on a similar subject.
Namely, Как вы себя... means how do you feel (as in emotions, or health-wise)
While Как вы чувствуете would mean how do you feel (how do you obtain your sensory information eg. by touching with fingers, hands,...)
There are lots of blogs online that talk about how startling and uncomfortable it is for Russians to be casually asked, How are you feeling? I recommend you read some Russians talking about it. In the United States, it is a casual greeting, to be answered with, "Fine, and you?" But in Russian it is a serious question. It can be considered rude.
I was looking up чувствовать/почувствовать (Imperfective/Perfective), and came across this sentence:
Я чувствую себя как рыба в воде = "I feel like a fish in water"
I guess that means "I feel like I'm where I'm supposed to be" or, more idiomatically, "I feel like I'm in my element" (element = place where I feel at home, natural, in a place best suited for me). In the US, we have a colloquialism, "He's as happy as a pig in sh*t", which means "He's really happy because he's in his native element."
I looked up the reflexive verb чувствоваться on Wiktionary, and the article said that this form of the verb is the passive voice of чувствовать, "in which the subject is not the person or thing doing the action, and is usually having the action done on them".
I don't know exactly how that would work with "to feel", but it would seem to make using чувствоваться a bit of a problem. Or maybe it's just that себя чувствовать is a preferred idiom.
Only in a certain sense. Neither has any semantic meaning. These are just structural elements of words.
The "-уете" is the ending for the second person plural form of a verb that ends with "-овать"/"-евать".
The "-ств-" is a suffix that is pretty common in forming words; for example: "чувство" ("feeling"), "правительство" ("government"), "действие" ("action"), "единственный" ("only"), "путешествовать" ("to travel") and many others.
The first "-в-" is simply a coincidence.
PS: you've wrote "здравствуйте", which doesn't end with "-вствуете", but with "-вствуйте". That's because it's an imperative. Essentially you tell someone "be healthy" as a greeting, instead of simply informing then that they are healthy.
But if we were to use the second person plural form it would've been "здравствуете".