No actual spiders, but the design and movement of one of the most common enemies is reminiscent of one (multiple jointed pointy legs, scuttling/scurrying motion)
Uncertain. Bailey Gilande, a side character, has a biography one can read, and in it, she is referred to with she/her pronouns but her gender is listed as X, and it is mentioned that she experienced some social challenges early in her career as a archivist. These challenges in question may be related to her gender, but as with all the character bios in this game, it is vague on a lot of details so it is open to interpretation.
The first body you stumble across in-game belongs to a character with tan skin (it's not really possible to tell his race), but he is just one of many casualties and not technically the first to "die" chronologically in the story and remains present throughout it regardless.