Sometimes. Although lymphedema can sometimes be painful, more often it is not. If you have a limb that looks like it is affected by lymphedema but it hurts, it is more likely to be lipidedema. Some say that lipidema is lymphedma with pain.
It can be. It can be, depending on speed of onset and degree of swelling, .
Can be. Lymphedema is when there is compromise of the lymphatic circulation. This third circulation (arteries, veins) carries things too thick to be safely carried in arteries and veins like bacteria eaten by white cells and digested food. It is pushed along by normal body movements. Blockage causes back up causing soft tissues/skin deformity which can be painful die to the tissues being stretched.