Here's where 歩けなく comes from:
Start with the verb 歩く, "to walk".
Turn it into its potential form: 歩ける, "able to walk".
Make it negative: 歩けない, "unable to walk".
Turn the newly formed i-adjective into ...
There are at least three types of omission of く, which should be distinguished.
The "traditional western" euphoric change is called ウ音便 and is described in this question, this one and a chart in this ...
It's basically that simple. So saying 寒くありません is more formal than saying 寒くない.
However, I feel that saying 寒くありません is a bit stiff even if you are trying to be polite. ...
ちゃう can be used in several completely different ways. Since you mentioned negation, I am certain you are entering the wild and wonderful world of the Osakan dialect.
As standard Japanese:
[Verb] + &...