"Kick the bucket" is a very slangy and coarse (грубый) way to say "to die". It would be cruel to use it in front of persons who are saddened by someone's death. Most people certainly would not use the phrase when referring to their spouse. "Slip away" means to die after one's health had gradually gotten worse. It is a gentle phrase, and people will not take offense if you use it under those circumstances. Also, although it is somewhat common in Russian to greet people you don't know with "Дорогие", in English greeting strangers with "Dears" sounds strange. Use "dear" for someone for whom you have tender feelings. However, you may also use "Dear" as a part of a salutation in a letter: e.g., "Dear Mr. Jones".

Yes, kicked the bucket would sound extremely rude. This expression is really only used as a joke such as "I slipped and fell yesterday, but luckily I didn't kick the bucket!"

I agree with the above answers. 'Kicked the bucket' is a very crude way of saying someone died and is not really in common usage (at least where I live). It is most prominently used in a humorous way to refer to your own death, for instance "if I kick the bucket doing that double shift tomorrow, you can have all of my stuff!" - or in some cases jokingly about the hypothetical death of a close friend. For example, after they do something stupid or dangerous, "how have you not managed to kick the bucket yet?" In my opinion, "slipped away" sounds very soft and poetic and would probably be used more in literary pieces. In common speech, I find that many people use the phrase "passed away" to refer to loved ones who have died. For example, "John passed away last year."