Is there a different way to say I live on the east side of the city? (which to mean means the eastern part)
Agree, my natural way to say this would be "I live on the east side of town" I am a native English speaker from the New York.
Doesn't 'I live in the east side of the town' work? Isn't town and city kinda the same?
Could we use "om" instead "av" here?
Would "district" also be accepted here? Or can this word be used in other senses for the word part?
No, it would not. District has some administrative purpose, while "part" is just a general term for a part of the city.
Could I use it like, "en del av tårta?"
Yes, that's a part of a cake then. Do note though, that a slice of cake is en tårtbit.
I wrote what Duo wants to see but I will keep using the east, west. south,north side of..!!