Veins do not connect to arteries. In the temporal area the veins and arteries can be very close together and right under the skin. Someone not experienced in treating (injecting) these veins could very easily inject an artery thinking that it was a vein. Injecting an artery usually results in loss of blood supply to the area that the arteries supplies and this could result in a serious complication. In experienced hands, and with the use of special venous lights and magnification and with proper knowledge of the anatomy, sclerotherapy in the temporal area can be done safely and with excellent results.
Arteries lead to arterioles then to capillaries where oxygen and nutrients are exchanges at the level of the tissue anad blood returns via venules, veins, and the big blue vein (vena cava) back to the right atrium of the heart.
Many people use lasers to treat facial veins rather than sclerotherapy. However, it's never as effective. The problem lies in each person's ability and understanding of anatomy, and injecting skill level. Facial veins can be extremely tiny - and so if a less experienced person injects, they can go right though the vein and inject the solution into tissue, which is not great. Personally, I've injected facial veins with sclerotherapy for over 25 years with extreme success and happy patients. But some doctors won't ever do this because their comfort level isn't there.
"This answer has been solicited without seeing this patient and cannot be held as true medical advice, but only opinion. Seek in-person treatment with a trained medical professional for appropriate care."
Veins and arteries do connect technically but through a tiny capillary system. In general treating large purple veins in the temples is safely and effectively done in my office with a laser rather than a sclerosing agent. Your dermatologist should be able to help you decide what treatment is best in his or her hands .