I just listened to an interview with a researcher of the common cold. Apparently, the average adult touches her face a great deal, including nose-picking, on average, five times per hour. All of us! Since then, I've been amazed to notice that I do it, too. Maybe not that often, but several times a day. Just gotta deal with an irritating boogie now and then, you know?
A person's own boogies won't make them sick (those germs are already in there), but they can introduce germs into the nose that were picked up from other people and surfaces. FREQUENT handwashing is the best defense, according to the research.
Your daughter has a habit, and is not alone in this. You don't, unfortunately, break habits by taking away toys or privileges. If there's an element of nervousness or boredom in that habit, punishing can actually make it worse.
My grandson was doing a great deal of picking for maybe a year, and all of us adults were getting on his case constantly. We just decided to quit. Ignored it completely. He stopped doing it except occasionally, usually when he's watching a video or going to sleep.
UPDATE: I just read your PS, and want to add that I have a great deal of sinus-related allergy. I've learned to carry a tissue, but two winters ago I went through a very bad stretch during which my nose was just plain irritated. I COULD NOT keep my fingers away – I was ALWAYS trying to fix/soothe the discomfort, and I worked my poor nostrils into bloody sores. What finally worked for me was a neti pot, or even just sniffing warm salt water directly from my hand, several times daily. After each salt sniff, I would rub a very sparing film of salve into my nostrils. It took weeks to heal completely. This might be hard to accomplish with your daughter.