the purple color comes from genetics and often environment such as cold - the spider is good - it will eat the smaller bugs which may be chewing on your leaves.. the speckles don't look too bad to me - here is one guess - if you water the plant and get its leaves wet in the strong mid day sun - the water drops can act like magnifiers and burn the surface of your fan leaves a bit.. there are lots of reasons.. it could be bugs or a plant infection, but it does not look very serious to me - if your adult plant comes out with a lot of these markings - then be concerned and post on here again - but, it looks to me like an isolated discoloration.. it is not covering the whole plant is it?