My wife was given a set of Noritake china that an unreligious friend of her’s received as a wedding gift back in 1984. The woman is Jewish, however, she is not observant in any way. She said that she opened the box of china, but never used it to serve any meals. My question is 1) can we trust that the china is unused, and kosher for use, and 2) do we have to perform Tevilat Kelim on the dishes?

If she was speaking plainly and naively, and the dishes look like they were never used you can use them. For Tevilah – you don't have to, to accommodate strict opinions you can Tovel them without a Bracha.