In addition to the films Roger Hicks mentions, Efke's emulsions have reportedly gone more-or-less unchanged since the 1940s or earlier. These might therefore be worth trying. Note that Efke's ISO 25 and 50 emulsions are both orthopanchromatic, which means they've got reduced red sensitivity compared to most other films. This will produce a distinctive look, which could be part of what you want. (I'm not sure when more modern panchromatic emulsions first appeared.)
As to film format, 35mm was certainly used for still photography in the 1930s. (The earliest Leicas came out well before then.) My understanding is that 35mm became much more popular a couple decades later, though.
My understanding is that RC paper is much more recent than the 1930s, so if you want to be as authentic as possible, you should go for FB paper exclusively. I'm afraid I have no suggestions for specific papers that are likely to produce the results you want, though.