Would laser resurfacing help with facial hair and or small veins on the face?

Thank you for your question. I suggest that you move forward and consult with your board certified facial plastic surgeon.
Best wishes,

Hello, thanks for your question.
Laser resurfacing, also known by its procedure name Pixel Fractional Laser, is great for improving skin texture, sun damage, pigmentation, acne scarring, scarring, skin rejuvenation, sun spots, skin tightening and wrinkles. 
However, when it comes to treating facial hair and small veins, laser resurfacing will not help. IPL would see better results.
Intense Pulsed Light (IPL) is a gentle, safe and cost effective way to permanently reduce unwanted hair on most parts of the body. It's also effective in minimising unwanted skin spots, leading to overall rejuvenated skin as well as a reduction in hair.
Regardless which you choose, a doctors setting is the best for treatments like these. Thank you and wishing you all the very best.
Stay safe and well, Dr. Murray from Absolute Cosmetic Medicine

Laser resurfacing is excellent for fine lines and wrinkles. Typically either CO2 or Erbium devices are used. These wavelengths do not offer much benefit for unwanted vascularity (blood vessels) or unwanted hair. There are, however, specific devices for these as well- I would recommend consulting with a physician who uses lasers on a regular basis to explore your options.