ACan I put the sps plates and legs in the dishwasher? Also, what about the Wilton plates? I don't want them to melt.

Usually yes, on the top rack.  Also once the cycle reaches the drying part unlatch the door and let them just sit to dry.

Some people think that by washing them in a dishwasher it will hasten them truning yellow.  Once they do yellow there is NO way to clean them up.  Many, many different tricks have been tried with little or no help.