"I hope you are doing well" does not demand a direct response.

You could just write, "Thanks for your e-mail", "Nice to hear from you", "How's it going?", etc.

I think you are close, captain.

Maybe: I'm doing well, thanks. I hope you are too. Yadda, Yadda.

#2 close to what?

Close to an appropriate response to the note Claus received.

You don't have to offer a direct response. That was my point. "I hope you are doing well" was just a blank standard formula to get the message started.

#2: Yadda, Yadda.

Don't include that bit, Claus! :-)

It's just a typical greeting expressing interest and concern. It just means something like 'Ich hoffe, es geht Ihnen/dir gut' or 'Ich hoffe, dass alles bei Ihnen/dir guht geht.'

I hope you are doing well.
I hope everything is okay with you / at your end.
I hope you're doing okay / all right.
How are things going with you?
How are you doing these days?

Thanks for your note. I'm fine, thank you, and you?
Yes, everything here is just fine, thanks.
Yes, we're busy but otherwise in good shape.
Well, apart from this awful cold I'm hanging in there, I guess.

That is: Even though a common reply is 'I'm fine, thank you,' we usually don't use 'fine' in questions; that would be a typical non-native mistake, just as 'I hope you are doing *fine' would not be very idiomatic. But questions with 'doing' or 'going' often have 'fine,' 'all right,' 'okay' in the answer.

Your answer 'I'm doing well' is of course also fine. (-: