What does it mean if the veins/tubes connected to the testicle are"swollen", have slight veiny lumps near the testicle?

Varicocele: That sounds like a varicocele, which is typically described as feeling like "a bag of worms". It is very common and generally not a problem unless it interferes with fertility. But it should be checked by your doctor to confirm.

If it is: On the left side it is most probably varicocele. I have no idea how old you are but if you are an adult i would not neglect this as sometimes "things" that happen in the kidney can cause the symptoms you have described. If you are young and have this for years i would just make sure your sperm count is good. But if all this are new , see your doctor! a urologist preferred!