Firstly, all pet insurance in the US requires you to first pay the bill and then submit it to be reimbursed. This allows you to go to any veterinarian of your choice. Also, it saves your veterinarian the extra work of having to contact pet insurance companies and work out the finances. They have enough to do already! :)
While pet insurance can cover a condition that your pet has had before, it does not cover pre-existing conditions. If that were the case, everyone with a sick or hurt pet would sign up and pet insurance would go out of business.
A few examples:
1. Your kitten had some diarrhea when you first got her. You took her to the vet and they figured out the problem and treated it. The diarrhea went away and she is fine. You sign her up for pet insurance. Three years later, your cat gets diarrhea again. Some pet insurance companies will refuse all claims for diarrhea because your cat has had it in the past, but with Trupanion for example, we would cover it since this is a new occurrence that is unrelated to the first one.
2. Your kitten has some diarrhea when you first got her. You quickly find a pet insurance company online and sign her up. In this case, they will not cover the costs of treatment because this condition is pre-existing.
I hope that this makes sense and I encourage you to call customer service for each pet insurance company and explain your situation so that they can help you understand what would and wouldn't be covered.