Yes, it will continue to work without internet. NAS and clouds are different things, a NAS is just a USB drive over the network, a cloud gives you more functionalities (versioning, backup, sync, collaborative folder, etc...).

What is exposed to the outside world (port forwarding) depends on your configuration. If you use SFTP, you will need to forward 22 towards your "local" cloud. If you store your cloud on the router, it will more likely be FTPS, and there will be no need for port forwarding.

You will need login/password/ip of the storage, and in the future when we implement E2EE, the password of your cloud.