You're allowed to lie on your application to the shelter. But you should also not let your cat outside because cats are bird murder machines.
A house visit to adopt a dog seems excessive. But this might be organizational scar tissue: mentally ill people adopting and hoarding animals, for example.
Depending on the jurisdiction, any adoption contract that says you have to allow unscheduled visits 6 months after may not be enforceable. Treat your adopted pet well and tell anyone who interferes to f off.