I just want to know can a Local Area Network broadcast domain be operated by only relying on APIPA?
Welcome to Network Engineering! If I understand your question, it's "can my network devices communicate on a single LAN using APIPA addresses?
The answer is YES. APIPA addresses are in the 169.254.0.0/16 range, so every device is in the same subnet, and therefore same broadcast domain. If fact, this is exactly what APIPA was created for: to allow hosts to communicate without the need for DHCP, individual addressing, etc.
If the computers are in the same LAN or VLAN, they will be able to communicate with the APIPA-assigned addresses. In fact, that is the reason that the APIPA even exists - to allow computers to communicate in a single LAN without requiring any manual IP configuration in the case DHCP is not available.
The only problem is the name-to-IP resolution. Usually, DNS is used for that but in a single LAN with APIPA addressing, the DNS would not be available. So you would either need to use IP addresses instead of computer names to communicate, or some other dynamic mechanisms would need to be used that allow stations to discover their names and IP addresses dynamically. Traditionally, Windows File and Printer Sharing services have such mechanisms. Other network services may not support it so it strongly depends on the kind of service you would like to use in an APIPA-addressed network whether you could use names or IP addresses when speaking to a different computer.
Please note that a switch never learns MAC addresses of connected stations into its MAC address table using ARP. This is a common misconception. A switch learns MAC addresses simply by observing the frames flowing through it. Click here