What is wrong in saying: my father doesn't walk
Just to clarify, this does not mean that पिता is not able to walk, but is not, in this moment, walking, correct?
Right, if you want to say he is not able to walk, it would be "मेरे पिता नहीं चल पाते" or more specifically "मेरे पिता पैदल(on-foot) नहीं चल पाते."
Is the rule of respect working in this sentence too? Subject and verb confuse me.
Yes, this rule is taking place. Since पिता is singular (father,) you know that any grammar points in the plural are to show respect = मेरे, रहे, हैं
Why nahin is coming before chal rahe hain? Shouldnt it be chal nahin rahe hain? Are both options possible?