If a test came back negative for strep, but I'm halfway through with a 10 day course of levaquin (levofloxacin) 500 (1 qd), should I discontinue now?

Yes. Most throat problems are viral and if culture was negative it is best to stop and not risk causing more problems by continuing.
Generally Yes. But you should rely on your doctor, not us online types, because there may circumstances unique to your case we do not know!
Yes. Antibiotics are not without risk. I would not take without reason. This way you avoid potential resistance selection, c diff colitis and possible side effects or medicine interactions (including: heart rhythm issues or tendon rupture).
Yes. Your sore throat is viral and and you now know finishing the antibiotic will not help you. In general we finish a course of antibiotics to decrease the development of resistant pathogenic organisms. Since you do not have any pathogenic organisms you can go ahead and stop the antibiotic now, and decrease your chances of the side effects of that antibiotic.