You need to be registered, but there is more to the answer than just that…
This depends on your settings. If you select the option that your page cannot be included in google search results then your profile won’t come up when people search for you…however, their is a caveat to that… you will still appear in google as a ‘friend’ of people you have friended. Try googling a friend of yours that you know has a FB account. Type their first and last name in quotes into google and check the results and you’ll see what I mean.
If you are logged in and search for someone that they aren’t ‘friends’ with, assuming their settings are ‘properly’ set you will only be able to see their profile pic, the network(s) they belong to and an option to ‘friend’ or message them. Some people don’t realize this or don’t care and you can view everything about them, peruse their pics, etc. That is risky.
She has to ‘accept’ them as a friend in order for them to view her page if her settings are done properly. She doesn’t have to know them per se, anyone can request anyone as a friend, it is up to the user to be judicious in their friend selection process.
Facebook has an embedded chat applet that allows you to chat with friends that are currently logged onto the site within IE, Firefox, Chrome, etc. So in essence these aren’t strangers… however… Your daughter can join any group she wants on the site (these group memberships are posted on her profile so you could see which ones). As part of the group their are discussion topics, wall posts, etc that she can be a part of and talk to strangers that way who ‘aren’t her friend’. Some groups are open to all, others are by invite only.
99% of people have a profile pic… a slight percentage less than that have many pictures of themselves on the site.
Personally I would worry more about your daughters friends on there posting something less than desirable about your daughter than her posting something. For instance… anyone can tag a ‘friend’ in a picture. If your daughter hangs out with friends at/after school, at a club, sport, etc and their are pictures taken they may end up on FB. Your daughter may not be fond of the pictures, but her friend could ‘tag’ her in it which means it is then linked back to your daughter and shows up as part of her profile. So for instance if you start a new page it will say ‘Jane Doe Pictures (1)’ then a ‘friend’ takes 10 pics of you at a party and puts them up on the site and tags you. The link on your page then says ‘Jane Doe Pictures (11)’. Granted you can untag the photos which reduces the count on your page and what your ‘friends’ can see of you, but those pictures are still on FB. Furthermore, when someone else posts pics of you, their friends and your friends can see them. If that user doesn’t have their settings restrictive, ANYONE can see the pics.
Some other tips: You can set up ‘access lists’ to prevent certain users that you are friends with from seeing your entire profile, pics, etc. So if you have 10 friends, you can have 5 see everything, your 3 family members only see 50% of it and your 2 co-workers see only your profile pic. If your daughter gets savy with the settings she can easily restrict her mom/dad via the ‘access list’ feature so you won’t know the extent of her usage and the content she posts. This is a very useful feature, but I am just bringing up the other side of it. It would be good to have her password as well.
If you have any other questions don’t hesitate to ask. Hope this helps.