Salutations are culturally freighted, and the most appropriate salutation to use depends on the locale, the social distance between the correspondent and the recipient, and the medium.
A style manual may be helpful when seeking consistency in such things as business correspondence, and in that spirit, I shall quote from an etiquette book:
“Honored Sir” and “Respected Sir” are now somewhat antiquated, but are occasionally used in addressing a person of advanced years, or by a poor person to a rich and powerful one, or in addressing a benefactor who has conferred great favors on the writer.
This passage is from Modern Manners and Social Forms by James B. Smiley, published in 1890. Yes, this phrasing was already considered old-fashioned in the U.S. 126 years ago. Americans will not encounter it today except in certain extremely prescribed protocols, as in some diplomatic greetings. Neither do I find it online in British or Australian sources since before the First World War.
Respected sir/madam (and honored sir/madam, and indeed respected and honored sir/madam) seem to be common in South Asia. As with the use of kindly (among others), it is a holdover from Victorian-era English. Possibly it has endured as a direct translation from the equivalent phrase in one or more of the subcontinent's other tongues. In any case, a 2014 New Indian Express column by Albert P'Rayan on the etiquette of salutations states
The salutations ‘Dear Respected Sir/Madam’, ‘Respected Sir/Madam’ and ‘Respected Sir’ are very common in Indian English. Senders of letters think that it is essential to address the recipient as ‘Respected Sir / Madam’ if the person is held in high regard or holds an important position. It is an old-fashioned term and native speakers of English do not use it. It is used in neither British English nor American English. It is good to avoid ‘respected’ in salutations.
I could not find any other authoritative guidance on Indian English greetings online, but if this columnist can be trusted, it seems respected sir, et al, may be common but falling out of fashion in India, whereas they fell out of fashion in Western Anglophone societies some decades ago.
Overly formal greetings, obsequiously polite expressions, grandiose humility, etc. may indeed have the opposite of their intended effect. Americans, say, or Australians may interpret suffusive politeness as insincere or patronizing, and take it with impatience or suspicion. Dear Sir or Dear Maam is sufficiently polite for business letters, and a personalized salutation (Dear Prof. Jones, Dear Dr. Smith) would be even better.
Finally, e-mail is a fairly new medium. The "rules" are much less firm than with traditional letter-writing, whether business or personal. As such, there is a huge range of treatments: sometimes I receive emails which are formatted literally as electronic versions of a paper letter— down to properly indented street address blocks— and sometimes they are one-liners that could easily have been SMS text messages.
As it happens, the most formal, most polite emails I receive tend to be from scammers. Ordinary business associates are more casual, and dispense not only with sir, but with dear — I myself tend to open client emails with hi, as in Hi Rachel or Hi Dr. Chen. Now, I am a technology consultant in the United States, and this level of familiarity may not be appropriate in other industries or in other parts of the world. But on the whole, the emails I write are far more informal than the equivalent I would put onto paper, regardless of my intimacy with the recipient.