This opening sentence is fairly standard, quite formal, and useful if you can't find a suitable opening sentence to start an email. Personally, I don't use it (my emails are quite short and direct). You can use it for someone you are communicating for the first time. An alternative is to just start with "Dear John, I'm writing to you in regards to …"

It's a safe greeting that can be used with anyone. I personally wouldn't use it with friends, but certainly with professors. It's a tiny bit more formal. It's not overly formal though, as someone already mentioned.

It's fine. It's not overly formal. It is always safer to err on the side of formality. This is a safe choice. You might well use it with a professor. You would use it with a friend if you hadn't contacted them in a while. In my case, to a friend, I might use a shorter, less grammatical version: "Hoping you're well. I thought of you today when I read an article about 'Tom Swifties...'" or "Hope all's well. We haven't got together in ages. What would you think about meeting for dinner at ...."

I like it! I don't think that it sounds formal. It sounds like you care. Could you tell me to whom you would use this sentence in your email? is wrong because the "use" should govern the "to whom", and it doesn't. So, starting again you could try with "Could you tell me to whom you would send this sentence?", which would be OK, but we don't talk about sending sentences to people, so we would have to change it to something that we send: mmmm, a greeting. "Could you tell me to whom you would send this greeting?". Better, but, despite what people say, people in these last fifty years don't use "to whom" much, so they say, "Could you tell me who you would send this greeting to?" or even "Could you tell me what kind of people you would send this greeting to?". Forget everything anyone ever told you about not ending a sentence with a preposition!

I am not sure whether I can use this sentence when writing emails to friends, professors etc. It sounds very formal. Hi Kailin, As you rightly pointed out, the tone of your writing depends on whether you are writing to friends or someone whom you know personally (informal tone) or to someone in an official manner (formal). It sounds more formal, but it is definitely a phrase commonly used in formal letters and emails. You can definitely use it to express concern and goodwill. I hope this helps. Cheers, Lance