In contrast to some of the other answers, let me suggest that, unless you are a peer of the other person, that you stick to the formal address until explicitly invited otherwise. Don't make any assumptions at all.
I always preferred that my undergraduate and masters level students addressed me formally, and never by first name. This was to emphasize that we weren't yet peers and their relationship to me was one of student and teacher.
On the other hand, with doctoral students (those already in the program) we, the faculty generally, tried to insist that students use first names. This was to emphasize that they were now "peers" in the sense of collaborative researchers, if not formal rank. Some students from more formal cultures resisted, but we were mostly successful.
But if you are a supplicant in some sense, such as in seeking an advisor, then the formal is always safe until the informal is invited, no matter how one signs their name.
For peers, on the other hand, these assumptions are generally benign.
Yes, I realize that I'm very old, but I'm also quite informal myself and times are changing. Nowadays youngsters might refer to parents by first name, though it would have been unthinkable when I was younger. Mom was always "mom", never "Liz", and for the same reasons as above. We weren't peers and emphasizing that can be important in learning.
So, call me "Dr. Dog" until I invite "Call me Buffy". And note that a lot of the people you will be addressing are closer to my age than yours. It isn't (just) a matter of respect for the person, but one of respect for the nature of the relationship.
I have no real advice on the "Dear..." aspect. Even for me it seems archaic, though I doubt that most people even notice it anymore since it is just a bit of boilerplate without much meaning.