It relates to the sound of final consonant. There are a voiced sound and a voiceless sound. (We separate this according to whether we use the vocal cords or not.) I have no idea this terms are right. You know, I’m Korean. For example, In 흔하다, 흔 (흐 + ㄴ) has a voiced sound of the final consonant. We say 흔치 않다, not 흔지 않다. In 섭섭하다, 섭섭 (서 + ㅂ) has a voiceless sound of the final consonant. We say 섭섭지 않다, not 섭섭치 않다. Of course, there are some exceptions, but I think this would be sufficient.

Oh that really makes sense! Thank you!

There's a similar shortening with adverbs: -하게 = -케

I'm clearly not a native and I'm not totally sure of this, but I think -치 않다 is a shortcut for -하지 않다. And it's only used with 하다 verbs. So, I think your last example is correct.

I think it means negative, like 안. I think you place it before whatever you are saying is not, make it present tense (치않아요), and tah dah! You have just said something is not something:) You should probably run this by a native speaker though...