Is 'én' necessary/preferred here? Could I say 'Az házam az váram'? Is 'Az én házam az én váram' more natural sounding to Hungarian ears?
It's idiomatic. "A man's house is his castle", that's how you put it in English. It means that people do whatever they want in their homes.
You could say "a házam a váram", but others might think you're incorrectly using the idiom.
I think you need the verb, "is", here, to make it correct in English and also to translate from the Hungarian where the third person [he, she, it] is just accepted as being there even though it is not written.