Usually it's okay to use the washing machine. Using the delicates cycle is an easy alternative to hand washing. Try it out on hosiery that isn't your favorite first. If...
Yes, but only wash with other fabric of the same texture and color, and preferably on the delicate cycle. DO NOT use any bleach or fabric softener in the washing...
No, wash your dyed shirt separately to prevent the colors bleeding into other clothes. After a couple of washes your dyed fabric can be washed with other clothes.