Do Americans often use "dear" and "best regards"? Yes, I see them sometimes. It really depends on the situation, though. They're very formal, so I don't use them when talking colleagues that I know well or are friendly with, for example.

I only use "dear" when I want to be extra, extra formal (like when I'm trying to get a job). Usually, to professors or to professional colleagues/boss, I say "Hi [Name]" or "Hello [Name]". As Torussan said, you can also just say their name.

Again, I only use "Best Regards" when I'm trying to be extra, extra formal. When I want to be somewhat formal, I say "Regards". If it's a person I often talk to, I sign off with my name only.

Dear or To to begin. Regards, Sincerely, or simply Thanks to conclude. 

Personally, I don't use either.

"Dear..." just sounds old-fashioned. In business writing, we are taught (university-level technical writing class) that the convention is to just use the person's last name as a salutation, followed by a hyphen: i.e. "Mr. Smith -" , not "Dear Mr. Smith -", so I don't use "Dear..." in a professional context.

For personal emails (I don't really write letters anymore), I usually just say "Hi" or some other greeting.

To conclude formal letters, especially when writing about an issue or problem, I prefer just say "Regards" (without the "Best"). It sounds more "distanced" and has a more neutral tone without the superlative. Otherwise, I would use "Sincerely" (for instance, if I'm writing to ask a question).

"Dear" is somewhat oftenly used when refering to a younger family member, usually by women, or also between people in a romantic relationship. "Best regards" is sometimes a closer on a letter or e-mail, but never really anywhere else.

Yes. I use them both. I live in Canada, but it's the same in the US.