When is it approriate to use actonyms and emoticons in business communication?

Emoticons, never.

The only time you may use acronyms in business communication is when they are business related. For example, "POS" is an acronym that means "point of sale" and refers to the cash register--retail management may use that acronym in business communication. Each type of business has its own jargon.

Acronyms like LOL and BTW are never to be used.

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The above answer suggests that an actonym is the same as an acronym, which is not always so.