Can I use a sterile needle to pop the little spider veins on my face?

No. Some activities, such as putting up drywall or painting, may be a better outlet for someone with a diy attitude. Medical treatment is best handled by a board certified physician. Facial spider veins are typically easily treated with a specialized laser or ipl see your local phlebologist or dermatologist. Put the needle down and back away slowly. :).
It won't help. Please don't try. It is potentially dangerous and won't help. There are far safer and more effective ways to treat them. These include laser, rf thermocoagulation and sclerotherapy.
No. No, it won't work. See a dermatologist or vein specialist who treats those. or veindirectory.Org.
Not safely. Please do not attempt procedures on yourself. Puncturing veins under the skin places you at risk for infection and scarring. Spider veins are treatable with sclerotherapy or a cutaneous laser.