"How are you?"
お元気ですか means 'Are you in good spirits?'
'Are you well?' works fine as an alternative reading. The standard answer to the question in Japanese is 元気です, which does mean (I) am well.
You can reply with just a happy はい or うん, or you can add those affirmatives to げんきです.
If you want to sound even more chipper, you can say うん！元気だよ！
'Are you ok?' would be different (大丈夫ですか？）.
'Do you need any help?' would require a far different and more formal greeting than this.
何かお探しですか（Are you looking for something in particular?) is the standard greeting in this scenario.
'How are you doing?' would be perfectly in line with the nuance of this question.
Japanese is a highly contextual language. Pronouns are often used only when it is absolutely necessary to clarify who is being referenced (which is why Japanese learning western languages struggle with them).
Additionally, Japanese usually refer to people by proper reference, instead of just saying 'he' or 'she' or even 'you'. For example, asking Mr. Tanaka if he is well it would be much more likely to say 'How is Mr. Tanaka?' instead of 'How are you?'.
Given the lack of anyone else being mentioned it is commonly understood among Japanese that the person being referenced is the other party (i.e. 'you'). Saying 'you' would be considered unnecessary and superfluous when not switching the topic of conversation.
I'd say "are you well/healthy" would probably be closer. Ok isn't exactly wrong but " Ok" is more often translated to 大丈夫 / daijoubu. With "are you ok" having a slight nuance of something recent happening and asking if they were unscathed, and "are you well" being their overall state of living/happiness.
You're more likely to ask if someone is ok if they were looking upset and you're showing concern for them; not really as a greeting.
Sometimes, sometimes not. Chinese and Japanese have both evolved separately since the time they first shared their writing system and Kanji went through a lot of crazy adaptations to fit the Japanese language. While Chinese has a single reading and meaning per hanzi, Japanese kanji have multiple readings and meanings.
The On-yomi (lit. 'sound reading') of a kanji is based on the original chinese reading (which may or may not be similar still today). In many cases these are used for their sound more than for their actual meanings though. Note that Chinese has far more sounds than Japanese too so this adaptation forces the original Chinese sounds to fit into the Japanese syllabary. So they may carrying the original Chinese reading but the meaning may be completely different. Since these are used purely for their sounds the kanji can be combined to create new native Japanese words that don't exist in Chinese. These are called ateji.
Kun-yomi is the Japanese 'meaning reading'. The sound is from the original Japanese word applied to a kanji with the appropriate meaning. So the meanings may be the same in both but they sound completely different.
Some examples I've seen come up in later lessons of both meaning and reading being different that has given some learners a laugh is 手紙 which means "letter" in Japanese and "toilet paper" in modern Chinese, and 聞 which means "listen" in Japanese and "smell" in Chinese.
Yes, but a bit closer to "Are you healthy?/ Are you lively?", it's an expression used roughly in the same way of "how are you". It'd be answered with 元気です - genki desu - "I'm well".
"are you alright" would be more like 大丈夫ですか daijoubu desu ka - sounds like you had an accident of some sort and someone is checking to see if you're okay
Language learning is more about finding a conversational equivalence than a literal word-for-word translation. Translating word-for-word will sound very awkward and sometimes isn't even possible.
Most likely not. お is an honorific prefix used most often with specific japanese (kun-yomi reading) words, with few exceptions ( 元気 being an exception)
学生 takes an On-yomi (Sino-japanese) reading and would require ご for an honorific instead.
Aside from that I don't think it would necessarily be wrong to add an honorific to 'student' but I've never heard it used yet so I'm not entirely sure if it would sound natural or not.