When someone has cancer in their lymph nodes, does that mean it's throughout their entire body?

No. Lymph nodes are "traps" that capture cell spread. Their involvement generally means that cancer cells have escaped from the source, but not necessarily that they have spread all over. Know that cancers of the lymph system (lymphoma) start there so in essence are not considered to have spread, unless they involve many areas.
Not necessarily... The lymphatic circulation is a conduit for cancer cells to exit their organ of origin and spread elsewhere in the body. Lymph nodes are filters which may "capture" some of these cancer cells; their removal then gives us information about what may be going on elsewhere in the body. The prognosis of lymph node-positive cancer is highly dependent on the type of cancer & it's chemosensitivity.