Let's look at each question specifically and expand upon your questions to help you better understand the situations that can occur.
1. Is Host A updating its own ARP table, or a shared ARP table amongst the entire LAN?
No, an ARP table is not shared among the entire LAN. Each device has it's own ARP table. It is the devices responsibility to manage their own ARP tables, including the local interface associated with the entry.
2. If the ARP able is not shared amongst all the LAN, would other hosts on the same LAN be made aware of this new information that Host A just received, if those other hosts did not have it already?
The short answer for a switched environment is no. If you view the format of an ARP Packet you will see that the packet includes both the sender hardware MAC address and IP. When an ARP request packet is sent from the requesting device, the packet is sent to the broadcast address and is forwarded by the switch to all interfaces (devices) on the LAN. This allows the device that has the requested IP Address to reply and target the reply packet to the specific IP and MAC Address of device that requested it. It is the network switches responsibility to maintain a MAC address table to forward the packet only to the interface on the switch of the specific device in the packet. Here is an example of an outgoing ARP packet from 10.0.0.1 in search of 10.0.0.2.
15:00:37.395072 ARP, Ethernet (len 6), IPv4 (len 4), Request who-has 10.0.0.2 tell 10.0.0.1, length 46
That being said, it is possible for devices to send out a gratuitous ARP reply packet announcing that they have the specific IP Address. The gratuitous ARP reply packet will be sent to the broadcast address, again, being forwarded to all devices on the LAN. In that situation, each device that receives that packet has to choose what to do with the packet. If they do not have the address in their ARP table, they usually ignore it. However, if they do have that address in their ARP table they should update any information that changed. Here is an example packet from 10.0.0.2 announcing to the LAN:
15:00:38.462135 ARP, Ethernet (len 6), IPv4 (len 4), Request who-has 10.0.0.2 (ff:ff:ff:ff:ff:ff) tell 10.0.0.2, length 46
All of the behavior detailed above prevents the devices from wasting processing cycles on packets that they do not need and memory on table entries that they do not need.