It depends if there is a DHCP server available. It it is truely an AP and depends on the primary router for DHCP service then your connection would show up as limited.
Sure, you can connect to an WIFI source that has no internet available. You should get a valid IP address if there is a DHCP server. A limited connection usually means that your computer tried to get an IP address via DHCP and did not get a response.
So if you try to connect to an AP (statically configured) SSID which is not connected to the rest of the network, this connection will be unsuccessful?
It depends if there is a DHCP server available. It it is truely an AP and depends on the primary router for DHCP service then your connection would show up as limited.