I don't wash them if it's going to be a gift. They look better not washed, and any new mother is going to wash it before putting on baby anyway :) Remember to put the soft facing on it when finished with the embroidery.
I don't wash anything before embroidery, but most of what I embroider is for customers and I don't charge for washing and drying in the price for my embroidery. I have had no complaints. If I am doing it for a gift, mostly towels I wash it after I have embroidered to get the solvy and chalk/pen marks out.
I wash it first, I wash/get wet and dry fabric before I sew it too....call me old school!
If it's for yourself or family, I always wash it first. For resale, I don't wash the items
Yes you should always wash anything you wish to embroider on or else the design will be distored after washing especially if it is something that tends to shrink.