Mom had a (painful) lump on her leg & now a lump on her arm. urgent care says varicose veins (in 2 sites?) should she get 2nd opinion? possibilities?

Yes: To be reassured a second opinion will remove fears of subcutaneous tumors or lyph nodes.

Varicose veins: This should be very easy to sort out with a physical exam & a vein ultrasound. If the MD at the urgent care suspects abnormal veins, then see a vascular surgeon for opinion (they are experts in varicose veins). The arm lump would be unlikely to be an abnormal vein (varicose veins don't form on the arm), but a quick exam & ultrasound should sort that out as well Good luck!