You can write them using a combination of ち and the small characters for 'y' as you already stated for cho, chu and cha:
As for che, you can write it using a small え:
I can't come up with any example word where ちぇ is used in hiragana. But the sound is used (and spelled in katakana) in some words, for example:
- チェック (check), where チ (chi) is used in combination with a small エ (e).
Would you be able to do the same thing with one of a, e, o, u, i. for example: ちぉ to make a cho sound.
As for your second question, this is what I think:
ちぁ,ちぅ and ちぉ are not correct, since the same sounds are already conveyed by using ちゃ, ちゅ and ちょ.
ちぃ does not make sense, because ち (chi) alone already includes the sound 'i'.
ちぇ is the only acceptable one. The reason for this is that there is no single hiragana character for 'ye', whereas there is one for 'ya', 'yu' and 'yo'.