It does use almost all vowels and consonants. “Ching” ㄔㄧㄥ is a technically valid syllable in the Pinyin system but does not correspond to any word in the dictionary.
No, it isn't; "ch" cannot be followed by a front vowel.
There are plenty of theoretically-valid but nonexistent syllables such as sěng or nǖ, but "ching" is not such a syllable.
And if you look up ch-/-ing, the table will tell you "no such syllable", just like it will for "m-/-iang" or "h-/-weng".
All three of those are invalid in terms of orthography -- the standard specifically defines that there is no way to spell them. But "miang" is perfectly valid conceptually, and "ching" isn't even that.
I know that biang is the name of a particular noodle; I'm not aware of kiang; and I believe that duang is a sound Jackie Chan made once. Which leads into a tangential question -- "duang" has no meaning. What would it mean to say it is "understandable" to millions of people?
[1] http://finance.eastmoney.com/a/202107151999911843.html [2] https://www.marieclaire.com.tw/lifestyle/art/58762