I have never seen "God bless" or something similar on a business letter, or even a personal letter for that matter.
In a personal letter, I've seen it used to say, "We will see each other this summer, God bless (or God willing)."
Here is a great article about closings
Very truly yours,
Respectfully,
Yours truly,
Sincerely yours,
Sincerely,
Best regards,
Regards,
Cordially,
With many thanks,
Warm wishes,
All of the complimentary closes above are perfectly acceptable, and choosing one is a matter of taste and fit. The first three are reserved for formal situations. The next, "Sincerely yours," is a bit formal, but it's also a standard close. Number 5, "Sincerely," is the most common close used. If you're not sure which close fits, choose "Sincerely."
"Best regards" has become more and more common, and it may soon eclipse "Sincerely" in popularity. "Regards" is the minimalist version of it. I don't like "Regards," since I prefer that people's regard for me be their "best" or at least "warm."
"Cordially" means "warmly" and "sincerely," but the word feels too reserved to me. I prefer "Warm wishes" or "Sincerely," which both feel warmer. But it's a question of preference--not appropriateness. I simply don't prefer "Cordially."
"With many thanks" has its proper place. However, the words "Thank you" are not a complimentary close--they are a sentence. They belong in the body of the letter with a period at the end.
The proper close for a letter of sympathy or condolence is one of these, or something similar:
With deepest sympathy,
With condolences,
In sympathy,
Sincerely yours,
As you have probably noticed, only the first word of the complimentary close is capitalized, despite the title of this post.
In business writing classes, people have asked, "Do I have to write Sincerely if I can't stand the person and I don't feel sincere?" The answer is yes. "Sincerely" is a much more gracious close than "Spitefully" or "With strong malice." (Note: I am joking. The two previous closes would never be used in a business letter.) Besides, "Sincerely" communicates positive energy and a knowledge of proper etiquette.
As for Madam... "Madam" is used only when we do not know the recipient's name, like this:
Dear Madam:
If we know the recipient's name, use one of these:
Dear Ms. Smith:
Dear Mrs. Smith:
Dear Miss Smith: