I have done this, but not just to shade it... I went to a much different colour. I liked the final effect; the printed white stays white, and only the background gets dyed, so it creates a very different look.
Guessing at answers to your questions:
(1) How dark it is depends on the amount of dye used, tone of dye used, and length of soaking time. I would do some samples (fat eighths, maybe?) to see how it works out. If you use a brown dye, it will probably be too dark; if you use a tan or lighter dye it's more likely to turn out how you want. Tea dying might be a better way to get the light shade you want.
(2) In my experience, Rit dye does fade over time. How long it will stay depends on how often you wash it. It's unlikely to fade away completely.
Good luck with it! I'm excited to see what you end up with.