So I brewed a light wheat beer, and once I rack it, there's a nice thick yeast cake on the bottom. Can I add a fruit juice, say apple juice, to the yeast and make a cider?
Sure can, I do it all the time. You may want to use only half of the yeast cake though, it can get pretty quick and violent on a whole yeast cake. Also, you may not think to use a blow off tube with a cider but I've seen whole yeast cakes make some crazy cider-krauseny stuff shoot airlocks off.
I did an experiment last year made 12 1 gallon cider batches with different yeast ...they all essentially tasted the same ...the difference was the speed they took to ferment...Us-04 will ferment cider in a few days