かぶり is an archaic word, and it's used almost exclusively in this idiom in modern Japanese. It's probably an example of a fossil word (an obsolete word that remains only in a certain idiom). かぶりをふる is a literary fixed phrase that only means "to deny/reject", and you cannot put another modifier like 横に in between. When the physical motion is ...
You can actually write it in both ways, and they will mean slightly different things.
過ち will imply error in moral judgement. 誤ち will imply accidental mistakes.
So it's better to write 過去の過ちを責めてはいけない rather than 過去の誤ちを責めてはいけない because in this case, you are not talking about accidental mistakes. Similary, it's better to write 計算を誤った than 計算を過った.
こき is the masu-stem of the verb こく, and こく should be in any dictionary.
扱く in Kotobank and jisho.org
放く in Kotobank and jisho.org
こき in 嘘こき is masu-stem as a noun.
Note that 扱く and 放く are different words although both are almost always written in hiragana.