Trust me on this, if people say they need time alone, then they need breathing space. You can only be part of someone's world if they let you, and a lot of people (me included) have a habit of fighting back when people try to enter against their will. It's scary when you tell people not to approach you and they approach anyway. As if you have zero control over what people do to you. As if you have no right for privacy.
She most likely would do it because she needs breathing space, and if she doesn't get it she WILL get pissed/moody/angry. xD
I keep giving you the idea I'm a social idiot >< sorry I wouldn't really do any of that but I mean, it's hard to talk about a theoretical situation where the only bit of information you're given about it is the picture.
As I know very well that everyone needs some personal space every now and then, I'd probably approach carefully, ask if everything was okay and if she wanted to be alone or if I could stay nearby, and if she wanted to talk or be silent. Then proceed appreciating her wishes. By no means would I enter the circle without permission, since it's so explicitly stated that it's forbidden. That would just be... rude. My girlfriend often needs personal space when she's anxious, but sometimes she'd like to cuddle away the bad feelings but can't get herself to tell me about it. That's why I often ask her if she'd rather have me stay away or hug her. It's good to talk about feelings before making sudden conclusions or jumping into action, that way I don't accidentally cause unnecessary extra anxiety to anyone.
I have no idea, I've always been like that. xD I might not try to invade her personal space too much if she says 'stop' simply, though. And awww, that's mean. <.< I probably wouldn't even try to do that to you, though. (I'mintimidatedbypeopleolderthanmeinrealife*cough)
I'd sit out of the circle and keep her company :v If she wants, she'll come out (and I may help her cut the leash), otherwise it's fine :v (I can't seem to comment/vote on your polls without getting problems apparently ò_ò