I'll be using Western examples as to avoid Chinese politics and to avoid my lack of knowledge of Chinese language 🙂.
I'll be using some characters that may not be correctly rendered in your computer if you lack a font that covers them. It is not a limitation of the Unicode or of this site HTML, but a limitation of your computer's fonts [a] (look at the end of this post).
These locales are region specific (representing mainland China, Hong Kong, Singapore and Taiwan, and this doesn’t even include Japan and Korea), but there still are multiple locales for each region.
Yes, but it doesn't matter character wise. All the same characters[b] are available in all countries.
[b] Well, technically, all the same codepoints are available in each of the encodings: UTF (all of them: 8,16,32,etc), GBK, GB2312, BIG5-HKSCS, EUC-TWUTF-8 and BIG5. That is similar to some difference(s) between Codepage 437 and Codepage 862 (to pick two random ones). Both share all the ASCII range 0-127 and each try to cover an specific language. Code page 437 is the character set of the original IBM PC (personal computer) and try to cover the generic use of Latin (not being good at any language). Code page 862 is a code page used under DOS for Hebrew. So, all GB2312 use the same numbers for the same codepoints and have available those same codepoints for any country.
Understand that all codepoints (and I do mean all) are available in UTF (any version).
The reason for having several countries is to deal with date formats, weekday names, currency name and all those other elements. Not characters.
That covers the first part of your title:
Which Chinese locales are needed to avoid missing ...
Pick any country you like (even US, GB, FR, etc. any) that use UTF (probably utf-8 in present days) and forget about missing characters, well, again, codepoints.
Note that I did not use the word characters, but images, the reason is long to explain:
Codepoints, as explained above are not the same as the images used to express a sound or idea. Human language is far more complex than that simplistic description. To have a working description of what a grapheme is we may cite Wikipedia (emphasis mine):
Grapheme In linguistics, a grapheme is the smallest unit of a writing system of any given language.1 An individual grapheme may or may not carry meaning by itself, and may or may not correspond to a single phoneme of the spoken language. Graphemes include alphabetic letters, typographic ligatures, Chinese characters, numerical digits, punctuation marks, and other individual symbols. A grapheme can also be construed as a graphical sign that independently represents a portion of linguistic material.
So: somewhat similar to saying: graphical symbol.
But the symbol might not be in a one-to-one relation to sound (in western scripts), as explained by:
The Word Burger
There is one sound for two
Nor is it connected to a single idea (meaning) as the image of the word
spoon, that is: Spoon may be connected to several ideas.
And Chinese is even more copious (in character count):
... the Chinese writing system is made up of an unlimited set of characters or logographs that represent a unit of meaning or morpheme (i.e., a word). Like any other language, Chinese has thousands of words. Thus, the Chinese writing system requires thousands of characters to represent each of its unique morphemes.
And each logograph (that represent a unit of meaning) may have several images.
Various glyphs representing the lower case letter "a"; they are allographs of the grapheme "a"
So, the answer to the second part of your title is inherently more complex than the simplistic "list of character numbers" (codepoint).
What seems to be a simple "R", may be a Latin character or a Mathematical symbol ( 𝐑, 𝑅, 𝑹, 𝗥 and 𝕽 are just some examples of math symbols) or even accented (Ŕ, Ř, Ȑ, Ȓ or Ɍ and more). So, in fact, there is not one simple "R", but some of them.
[a]To be able to present all this different codepoints you need fonts.
Fonts have a concept called coverage. How many codepoints are included.
An extreme (very big file) example is the coverage of all (visible) codepoints in the BMP by Unifont.
For example, 𑄌 (U+1110C CHAKMA LETTER CAA)Chakma Alphabet is not a letter you plan on seeing but it is included in Unifont.
There are other open (free) fonts with excellent coverage, like Code2000
An example of different fonts in Chinese is in this wikipedia web page