Oh, and "dos" as plural can be contradicted with what Chris B pointed out: "there are two Is and two Us in the word ridiculous". You see, it can be argued for both sides. "Dos" as plural needs to be re-written in the dictionary because it lists "do's" as a noun. Someone got confused at some point.

Capitalizing both words is incorrect grammar. It doesn't matter if it looks more pleasing to the eye, it's still incorrect. Even for titles and headings, only the first letter of each word should be capitalized.

This is one of those special case scenarios that exact definitions fail to cover, ergo such confusion. Let's resolve this in a simple way that doesn't involve textbook definitions that somehow still leave us unsure:

For English, "dos" is not a word, so the apostrophe is needed.

Don't = do not. "Do nots" is incorrect - nots is not a word, so the apostrophe is also needed.

Just because people have always been oblivious to use "don't's" doesn't mean it's not proper use of grammar.

Do's and Don't's = correct
'80's = correct