Can lymphedema spread from my arms to other parts of my body?

Yes. Upper extremity lymphedema most commonly occurs following the treatment of breast cancer, which involves lymph node removal +/- radiation. The entire lymphatic system is interconnected and swelling can occur anywhere in the body regardless of which lymphatic region was affected. Though not curable, this does respond to physical therapy.
No. Lymphedema is not an infection and typically does not spread. However if the lymphatics from one leg for example become overwhelmed they may cause swelling on the other side. See a vascular surgeon who specializes in lymphedema and get appropriate care.