Often, chest veins are isues in people who have thin, white skin as the green blue hue of the varicose veins are quite obvious. These are very effectively treated with one or more sessions of foam sclerotherapy with high succes rates. Sclerotherapy can also be used for breast veins.
These are superficial and cosmetic veins and treating them will have no untoward consequences and will NOT cause problems with arteries as you have suggested.
Laser resurfacing, laser for vein treatments on the face or hemangiomas or capillaries do not harm facial nerves. There are motor nerves that are quite deep aad definitely not affected. Seensory nerves are also not injured with laser.