I have washed 1/4 yd (not FQ) cuts. I wouldn't wash anything smaller than that. But I typically don't use pre-cuts, other than fat quarters.
I just bought my first 2 1/2" strips for a pattern which calls for 27 unique strips. Since my (small but growing) stash is all pre-washed, I purchased 1/4 yd cuts for the extra pieces needed to have the full complement of unique strips. I won't wash any of it until the top is completed.
I agree - don't mix washed and unwashed.