What are possible causes for a one sided large, soft, non tender swelling palpable under the jaw or in the back of the throat? no fever, rash, lesion

Several things. . .: If you are talking about swelling under the jaw: salivary gland stone, salivary gland tumor, inflammatory lymph node, cyst, lipoma. I am not sure what you mean by the back of the throat but possible swellings back there would include: tonsil edema, tonsil inflammation, fluid collection due to an abscess (but there should be pain) or noninflammatory causes.

Not tumor: In addition to dr. Zapanta's explanations, would comment that tumors are hard, not soft (lymphoma typically ruber-hard, carcinoma just hard). Inflammation secondary to infection may be the most common cause if its tender...