Moving with a cat

The best thing to do: go to your local vet. He/she will be able to inform you on all things you need to do to migrate your cat to Switzerland.
When looking for accomodation, mention that you have a pet. The standard formulation in rent contracts is that "pets are not allowed unless permission is granted". So: ask for that permission. Shouldn't be too much a problem as long as your cat doesn't annoy your neighbours.

Switzerland is a well-organised country and "contract is contract". If the contract mentions that you need permission for a pet, *do* ask for it (on paper). Having no permission could be a reason for a landlord to kick you out. This may not only happen with "bad" landlords, but also in a situation when the appartment block is sold to a new owner. You never know how your new landlord will be and once you've got the permission, you've got it.