Personally I wouldn't have a problem with it. But if you do have trouble pulling cord, you could strain something which is not what you want to do at 32 weeks. Maybe someone could get it started for you if that's the case. I'm it could be good exercise as well.
My husband has a Kubota tractor (we have 6acres) so he taught me how to operate it in case I need to mow the lawn for some reason. I drive it occassionally, but so far haven't had to cut the grass.
I wouldn't because of the fumes...
I don't know if I would be using a pull cord mower, if it was one you ride I wouldn't see why you couldn't. If it's one that's easy to start without having to pull really hard then I think it would be ok, but if it takes a lot to start, maybe ask a neighbor to start it for you?