If I write to my friends I don't bother addressing them or signing off (since I'm younger it seems way too formal, but for older people I'm sure it sounds normal). I might just start off with "Hey," and that's it. With close family I might be slightly more formal and sign off with just my first name, but it's not very often that I email family.
When I email a teacher I'm more formal, but I don't start off with "Dear Sir," I just start with their name, "Mrs So-and-So," and sign off with "Thanks," followed by my full name. I don't tend to ask my teachers personal questions like how they're doing, because I find that a bit weird, but maybe not with coworkers or classmates.
If you're going to ask someone a personal question I'd put it at the start, then move onto the professional stuff. Asking someone how they are is more of a conversation starter than anything.