Well the answer turns out to be no you can't. Some of the story missions actually require you to be a part of a fire-team of other players, so no xbox live gold, no completing the story missions either.
Just found this out now as i've got to level 20.
I'm not aware of an official lore answer on any of this, however:
The Eliksni ('Fallen,' to most of humanity) are insectoid/arachidlike in biology (https://www.destinypedia.com/Fallen#Biology).
Much of their physiology seems to be more spiderlike, in particular, but this is already into conjecture as Earth spiders and the Eliksni have no common biological ...