Assalamu alaikum
I think that the veil is arabian cultural influence within Islam. I read a book about 4 years ago or so that touched on the subject of women's dress during the time of the Prophet, and apparantly most women veiled anyway, and dressed in the style of Beouin women. A scarf type garment was pretty much a common sight, no matter what religion the woman followed.
I would go so far as to say that if Allah choose a swedish person as our Prophet, and revealed the Quran to him, we wouldnt even have niqab. etc now as the cultural style of dress in Sweden at that time was totally different to Arabia's. ( Hey..we could even be wearing horned hats now ;D)
I personall dont think that a scarf on the head is required in Islam. The Quran tells us to cover our chests, be modest and lengthen our garments. How some scholar came to the conclusion that these words mean women must cover everything except two, or even one eye I will never know.
I wear a large shawl type cloth over my head when I pray, but not because I think I have to. It sort of focuses me in my prayers and shuts out the world, if you can understand my meaning there.
I certainly dont believe that Allah wishes every woman to walk around in head to toe black robes with only there eyes showing. The Quran tells us that Allah has made our religion easy for us, but I dont find it easy wearing these clothes ,going about my daily business in the heat.