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As far as I know, there are only two Simsun fonts, Simsun Regular and Simsun-ExtB Regular. MS Word can style both as Bold, Italic, and Bold Italic, but ID reports these as errors when it finds them.
Meanwile, the missing chars. could well be a matter of applying Simsun Regular to characters encoded in Unicode's CJK Extension B up in Unicode's 2nd plane. If so, applying Simsun-ExtB Regular should do the trick. A handy MS Word trick for checking a character's encoding in Windows is to select it and hit Alt-x: if that yields five hexadecimal alphanumerics instead of four you're up in the 2nd plane.
MS Word fakes italics by slanting type. ID can too, but you have to insist, most conveniently via a char. style. MS Word's artificial bold is similar, and again you can force InDesign to make chars. look bold. But it would be much easier, not to mention more attractive, to use a heavier weight. As I say, Simsun offers just one, but ID bundles several weights of Kozuko Mincho Pro (with the usual caveats about using Japanese kanji for Chinese). Or you could wander down the hall to Adobe's CJK Type blog and see about downloading a suitable weight from the open-source collection, Source Han Serif. These fonts offer 7 weights, each in several local variants. Note, however, that Source Han Serif includes just 65,535 glyphs, so it can't cover CJK Extensions later than A.
Thank you a lot for replying. I am quite new to InDesign, so what you said was pretty confusing. I replaced SimSun Bold, Italic, and Bold Italic with SimSun-ExtB, the rectangles still appeared with red highlight. The Adobe's CJK Type blog is even more confusing to me, sorry. Can you please explain it to me in an easier and clearer way?