Here's a great place to check out various filters and their effects on your image. https://tiffen.com/diffusion/
I don't think mimicking the DOF will look inauthentic. As long as the viewing format is the same (like 16:9, for example) as it would be otherwise, no one would be able to tell based only on DOF. It will be other things that give it away.
While in my experience it is harder to get shallow DOF shots with 16mm, it is possible, and I did it fairly often in my recent film (I shot on 16mm Kodak Vision3 film): https://www.youtube.com/watch?v=dLOf-0GoV5E&t=26s
I think that one of the biggest things you can do is to use older lenses, filters (such as Black Diffusion FX, Black Promist, Soft, Glimmerglass, Pearlescent, etc.) and some post-processing (adding some grain in post) to get a film-like look, in addition to the kind of lighting and shots you use.
Something worth trying is the film emulation plugin FilmConvert. I use the Premiere Pro version and it does a good job of emulating various film stocks. One of the things is does when it emulates 16mm is actually lessen the sharpness of the image a good amount, as well as introduce film grain, etc.
That is incorrect. Your shutter speed SHOULD be DOUBLE your frame rate.