is internet access for the appliance mandatory or can I drive the EBA without internet access?
No, its air-gapped ready. However, to run your functions, it will need connectivity to the registry where the containers are hosted whether that is within your datacenter or public registry like docker or gcr
which private registry do you recommend for inexperienced users?
VMware Harbor, its free/OSS and its also what we recommend if you don't have a solution in place already. If you need commercial support, consider purchasing TAP or TKO