I think this is a variant of the /ai/-to-/ee/ sound change that typically happens with i-adjectives:
What does こまけー mean?
What is じゃねぇか? What is its original form?
わからない vs わかね in My Boss My Hero
If I understand correctly, in the past, ちげぇ (also written as ちげー or ちげえ) happened only when ちがう conjugated to the continuative form, ちがい (chigai). For example, ...
It's a colloquial, contracted pronunciation of
"If you leave what you (are supposed to) leave"
It consists of:
置いていく(置いて行く) ⇒ 置いてく
もの(物) ⇒ もん
を is left out
置いていけば(置いて行けば) ⇒ 置いてきゃ
置いてく is a relative clause that modifies もん.
［置いてく］もん literally means "things [that you leave]".
First, yes, てしまう is commonly contracted to ちゃう (and でしまう to じゃう). They mean the same, but ちゃう is more casual as a contracted form.
Second, as for the meaning of しまう・ちゃう here, I think the slightly less common usage of "do completely" fits, with ちゃって being itself the て-form of ちゃう, making it a casual/friendly request. This usage is very common, and is derived ...
てやんでえ used to be a Kanto dialect (nowadays it's more of a stereotypical phrase associated with edo-period commoners, and is not actually used in daily contexts). It comes from 言っていやがるんだ and translates to "what the hell are they/you saying". It's NOT a Kansai or Kyushu dialect. For example:
This 期待されとる is short of 期待されておる (see this chart), and in this context おる is the same as いる for progressive aspect. なんぼ is the equivalent of いくら in Kansai/Shikoku dialect. や is widely used instead of だ in western Japan.
= No matter how much you are expected (No matter how much they expect on you), if ...
歩けん is a variant of 歩けない ("I can't walk"). Usually a small girl doesn't use this form because it sounds a little old-fashioned or dialectal. Maybe the speaker is an old vampire or someone playing the role of it?
Is verb ending ない shortened to ん?
Verb conjugations such as 思わん、言えん
The omission of を is very common in colloquial Japanese. There is ...
It's almost certainly a contraction of 見てはいない ("[someone] is not seeing it" or "[someone] has not seen it"). It can also be a contraction of 見てしまいなさい ("Watch it anyway!") at least in one western dialect I know, but it's rare.
In casual speech, ては very often contracts to ちゃ (and では contracts to じゃ). See Purpose of adj+kucha and ...
I am sure that you will sometimes find the expression 見ちゃいない in a manga, because this expression is very colloquial and used in daily conversation. I think the following expressions can arranged in the order of formality. 見てはいない、見ていない、見てない、見ちゃいない。
かっかといで is just a contraction of かかっておいで "Come at me!". (The contraction of て＋お > と is the same one that occurs in -ておく > -とく.)
As you may know, おいで is a fossilised form of the honorific of くる (おいでになる), used exclusively as an imperative. So essentially, this is just equivalent to かかってこい.