That sounds weird to me. I've heard "ethnic restaurant", meaning one that specializes in the food from any certain country, if you're speaking about those kinds of restaurants generally. It depends on the context. If you're talking about a restaurant in your own country that specializes in local food, you could say "local restaurant specializing in ___."

It is grammatically correct. The sentence is strange in its meaning. We need to know whose heritage. Italian heritage, Asian heritage, or some other place name. I think a better way to say this might be to use the word 'cuisine.' I went to a restaurant that had Italian cuisine. I went to a Swedish cuisine restaurant.

It sounds correct, though I don't know what a heritage restaurant is. Aside from that, the sentence seems correct.

"Traditional" is the word I would normally use, if I understand what you mean. You could also use it in conjunction with the type of cuisine you are eating: - A traditional Chinese restaurant - A traditional Italian restaurant Meaning it has the decor, food quality and feel of the country the cuisine is based on.

Yes,It's correct