"Peter's mother is not happy."
Translation:पीटर की माँ ख़ुश नहीं हैं।
You would use की because it refers to a female noun(मॉं). Even when you use the respectful form where मॉं is treated as plural, it would still remain a female noun. Compare with पीटर की बहनें खुश नहीं हैं।(Peter's sisters are not happy).
के is used for plural and male(or mixed) nouns. For example, 'Peter's father is not happy' would be पीटर के पिता खुश नहीं हैं।
के (well, masculine-plural form) is used as an honorific in Punjabi. I wonder if some dialects or speakers of Hindi do the same. It seems plausible since "Punjabi" shades off into Khari Boli "Hindi" of Delhi, and many Urdu speakers of Pakistan were migrants to Punjab. I wonder if PrajitDhar has some experience with this sort of usage.
nahin comes before the verb, in the standard word order.
It would help to know where you saw nahin before an adjective.
Possibly, the adjective is linked to a verb in a phrase, like garam-karna (warm-do, "to warm, to make warm") and someone treated it all as one big "verb phrase", and stuck nahin before it.
Other possibility is a switch in word order for emphasis.
But if you want a rule, it's that nahin comes before a verb.