Why do we Veil?.
What does it mean to Veil?
1. Personally, I am uncomfortable with the term "to veil." The Church has never required women "to veil." Before the updating to the Code of Canon Law in 1983, the Church required women to wear head covering in church. That could be a veil, but in the United States it was more commonly a hat. These days, women who want to cover their head can do so with any appropriate head covering they like (e.g., hat, scarf, veil, etc.).
2. Those who choose to wear head covering give different reasons for doing so. Some do so for spiritual reasons, such as a sign of respect for our Lord in the Blessed Sacrament. Others do it for cultural reasons. In my own case, I do so when it is expected by a community I am visiting (such as the FSSP church in my diocese). I don't do it for any spiritual reason, but I respect the choice of other women to do so for their own spiritual reasons—and I hope they respect the freedom of other women to make a different choice than they've made.
3. By "anyone," I assume you mean any woman attending a Catholic Mass. If so, yes. Any woman attending a Catholic Mass is free to wear head covering. Men, on the other hand, ordinarily remove hats or other head covering (e.g., hoods, caps) while in church. (Of course, if a man has a medical need to wear head covering, such as to keep his head warm during cancer treatments, he's free to wear head covering.)
4. Those who choose to wear veils are free to wear any color of veil they prefer. One of the reasons I am rather uncomfortable with the surge in veil proselytism is that it can make Catholic women fear that not only must they wear something on their head but that what they choose to wear will be somehow inadequate. I would caution anyone who encourages "veiling" to refrain from anything that creates false scruples in others.
5. No. If a young girl wants to wear a veil, and her parents are okay with her doing so, that is a personal family choice.