I would definitely leave out the 'doing' if it is a formal letter, which it looks to be

@Sarah - not necessarily. How do you know that?

@Sarah: Nur interessehalber: Ist es wirklich so, dass in formellen Briefen das "doing" nichts zu suchen hat?

"I hope you are fine" wollte ich nicht benutzen.

Also müsste "I hope you are doing well" gehen oder? Wenn ich dies benutzen würde, wäre es ja nicht in geschäftlichen und/oder öffentlichen Briefen.

to my ears, "i hope you are well" sounds better than "i hope you are doing well", whether formal or informal.

there's not very much difference between them. the first is mainly about wishing someone good health. the second is possibly a little more about hoping they're "getting along" ok, that their business is profitable, they've found a mate, and so on.

it's difficult to explain why, but "i hope you are well" seems more safe. unless you really mean to say something like "i hope you are doing well in your new job", or "i hope you are doing well in california" (found a job, got an apartment, etc.)

"i hope you are fine" really doesn't sound right.

I agree with Sarah about leaving out 'doing', and also I think she has a point about a formal letter. That was my initial reaction too, because this sort of expression is fairly formal and complaisant - it is what you would write in a formal letter to an older or more senior person, whereas in an informal letter you would say something like 'hope you're okay'.

to a friend, i would write:

hi -

how's it going?
how are things?
hope you're well...
i hope everything's going well.
etc.

f.y.i., "hope you're okay" or "hope you're alright" is something i'd say if i heard that my friend had been in a car accident or something, to me it means "hope you're not hurt" or something.