Dora - as others have said, a red line in the playground is a way of keeping the children in 'their' area of the playground, without putting in physical barriers. The dses old primary school did this, to keep the infant and junior children apart - so the bigger juniors weren't running through the little ones, risking knocking them over.
Regarding the teacher playing with your child - this was, I am sure, the teacher getting to know her new pupils by playing games with them - a very normal and good thing for her to do.
I agree that you should go back to your gp and talk to him - and get him to chase up the referral for you. He might be able to arrange for you to see a community psychiatric nurse in the meantime, who may be able to help you put your fears and anxieties into perspective.
Something I have found has helped me, when I have had these horrible worries that won't go away, is to do this visualisation. Visualise yourself packing the anxiety/worry into a box that is attached to a big balloon. Close the box and then let go - and watch the balloon carry the worries away - seeing them get smaller and smaller, and eventually vanish altogether. Feel the weight of the worry leave your hands as you let go of the balloon.
I hope this helps a bit, and I hope you can get past these worries and be happy.
{{{hugs}}}