-chi, -cchi, -chin are informal honorifics. They are a deliberate mispronunciations of "-chan", typically used by teenage boys/girls trying to be cute. Since most teenage girls want to be considered cute, they will tolerate this from close friends. They are used to build nicknames or use it to playfully tease/insult. Some also use this suffix to the informally name people that are cute and famous i.e. pop idols. Apart from food wars, I have seen this suffix being used by Yasuhiro Hagakure from Danganronpa franchise.
Coming to -nyan, it is pretty self-explanatory. It is used to express joy, satisfaction, cuteness. It is used mostly in anime. However, in this case dialect have somewhat of a relationship. Due to a specific dialect, a village in Japan end their sentence with "-nya/nyan". But in other places, you'll never see it being used apart from maid cafes (because it sounds cringy(?) [citation needed :/]).
Source:
What are the differences between Japanese honorifics chan, chi, chin and in? - Quora post
Japanese Names and Honorifics in Anime
https://gamefaqs.gamespot.com/boards/945498-shin-megami-tensei-persona-4/44399981