I believe the mincho font (pretty much the de facto standard of modern kanji) does have italics. They're not used all that often or to the same effect as in English though.
Others have also pointed out the dots which are quite common in formally printed materials. In emails and such, I see people use the 「」 quotation bracings as emphasis from time to time.
Dakuten(what you call “dots over letters”) are used to modify the way consonants are pronounced, not for emphasis, e.g to turn the character for “Ka”(カ) into “Ga”(ガ).
And the usage of katakana in the way the parent describes isn’t “tiring.” It follows almost the exact same pattern as words are italicized in English, where a word or phrase is represented differently from the surrounding text. Less common/standardized, but not tiring.
EDIT:
My bad on the bouten/dakuten confusion. Apologies.
As @chch mentions, the parent probably meant bouten/etc. I couldn't find an English article on it, but the Japanese Wikipedia page has examples and some cool CSS styling info that's related:
As for katakana, it really does depend. For common or short words, people usually don't have a problem, but if you throw a whole katakana sentence at a Japanese person they will often stumble and reading speed noticably decreases.
Their point about katakana was also correct - normally a word is either written in katakana or it isn't, regardless of emphasis. Using katakana for emphasis happens but it's not common, and would get tiresome very quickly.
It does, however, have katakana -- which is sometimes used in a similar fashion.