Pets and smoke are both common triggers of asthma. If you think the school isn't providing a suitable environment for your child, talk to the headteacher about it or, if you have a Parent Teacher Association, you could bring the issue up there. However, your child may encounter difficulties while at school which you need to be aware of. Asthma can be triggered by things which are a regular part of school life - wood dust, tippex, chemical fumes, deodorant spray for example. Let the teacher know if your child is having trouble with a particular class - and try to work around it.
Exercise is good for everyone, including children with asthma.
Even if exercise is one of your c0hild's triggers it's important that they continue to keep fit and participate in as much sport as possible. At these times their airways will be extra sensitive. Ensure that your child has access to their reliever inhaler at break times and make sure that they have a warm coat in the winter. If your child does need to stay indoors at any time, come to an agreement with staff about indoor supervision. From time to time your child may not be well enough to attend school.