I would not be happy with this school at all. Reception children can sing and chat during play time and other children shoul NOT be saying shut up - if they are annoyed they can move away, or maybe ask if she can use a quieter indoor voice (at most) - it's play time for goodness sake! My daughter is also shy but the school know this and they've really helped her develop and she has come out of her shell so much due to them taking the time to get to know every child.
was this school your first choice? Did you get a clear impression of what the school ethos was and what their main school rules are?
I think S&L therapy going in to schoolis a great idea.
Btw I have times even now as an adult where I can't make eye contact and it was worse as a child. I am not autistic or aspergers but I DO have a lot of anxiety. For some people it can be hard to spot the difference.
However even if she was autistic it wouldn't be recommended to tell her to shut-up when singing, so that's pretty unacceptable in my view.
Ps I wouldn't worry about the dyslexia just now. At this age so many children have the classic symptoms and most reading and writing schemes adopted by reception classes are multi sensory (which is the best approach for dyslexia anyway and helps all children that age). As she gets older you'll be able to tell if her symptoms persist or follow the more usual pattern (my daughter currently confuses b and d, u and n, writes backwards sometimes and has just started inserting random capitals. She could be dyslexic like her dad but these are also the normal stages of reading and writing, so for now I'm just watching and waiting).