I actually wore a cathedral veil with a blusher (this is what the face covering is called). We had a Catholic Mass and I chose to wear the blusher. It's not uncommon for some churches to request that the bride has a headcovering, so yes, some "modern" religious brides still do wear veils. In secular weddings, it's less common, but some brides still choose to wear one.
Our priest informed us that the blusher was to be lifted after my now husband and I walked down the aisle but before we were seated (the couple sits during Mass ceremonies). So in my case, my husband lifted the veil since we walked together. I was not "given away".
So really I only had it down for the 45 second walk down the aisle before the ceremony began. Good thing too, it was hot! And awkward. :) It did make for some stunning pictures though, I'm not going to lie.
I think that nowadays it's common for the veil is to be lifted before the ceremony begins, as it was in my case. Who lifts it depends on your ceremony structure. I walked with my now husband, so he was the one to lift the veil. Same would likely be the case if you walk alone. Other brides have their father or mother (or both) do it. I'm sure a bride out there has also lifted her own veil. Unless your church (if religious) or officiant requests a specific way, the choice is determined largely by your entrance.
As for the decision to wear a veil, there are so many factors that really only you can decide. Some brides wear one for religious reasons, others believe they "represent" something, some think it's "traditional", others simply like the look. Conversely, some women hate veils for the same reasons.
Have a wonderful wedding.