Is having a lymphangioma painful to the child?

Not usually. Lymphangiomas contain no nerves, and thus the size and stretch caused by one does not cause pain. However, a large one overlying a body part that is needed a lot (eg, the eye) can be really annoying. More than anything, they are a cosmetic issue, and if they are hindering daily function or causing great distress, they can be surgically removed.
Yes. Lymphangiomas, or more correctly, lymphatic malformations result from abnormal formation of the lymphatic system & are characterized by obstruction to lymph flow. If they become infected or enlarge in tight anatomic spaces (i.e., neck, extremities) they are painful.
Not necessarily. A lymphangioma is a malformation of the lymphatic system. It may be an obvious deformity of the head and neck area, axilla/chest, and or groin area. They can get infected, but if large look worse than they feel. They can be treated by surgical removal.