Can I related the crash to availability mode settings?
No. By the looks of your log messages, you actually lost cluster quorum due to the removal of two nodes' votes, and then your witness (file share). Is this a 3 node cluster with a file share witness perchance? In that case, if you pulled these events from one of the node's event logs, then it may appear to each of the nodes that there is a lack of communication with all voters. That would generate a similar, if not same, error footprint like you have above. Nobody can talk to anybody, if that is the case.
During that time, the quorum will be lost as you are currently seeing. There is a level of assuming here, as I'd need to see way more diagnostic information to pinpoint the cause of voter removal, but that is why the quorum was lost.
Regardless, this appears to be a problem that surfaced in a down cluster, in which case your availability mode would have nothing to do with the WSFC cluster failing.
As for "best practices" for the availability mode to go with, you need to determine requirements for data loss, performance impact, and a few other factors that are best described in this BOL reference on Availability Modes.
Does the Windows Failover Cluster for a multi-subnet SQL Server
Availability Group require a static IP entry for each subnet?
The CNO will require an IP address for every subnet it could reside in.
I am running SQL Server 2012 on Windows Server 2012 Hyper V VMs in 2
separate subnets in the same domain. I understand that I will need an
IP from each subnet when I create the listener for my AAG. What I am
unclear on is the configuration of IPs on the underlying Windows
Failover Cluster.
For the underlying WSFC you'll need at a minimum:
Node1 - IP Address for each unique subnet for each network interface
Node2 - IP Address for each unique subnet for each network interface
CNO - IP Address for each unique subnet
EX: 2 nodes, 2 subnets, 1 interface per node, subnets 192.168.1.1/24 and 192.168.2.1/24
Node1: 192.168.1.10
Node2: 192.168.2.10
CNO: 192.168.1.20, 192.168.2.20
Also, if the server hosting the secondary replica does require its own
IP, does it also require its own unique cluster name (and can you
explain why this is necessary)?
I'm not sure I understand this part of the question. All of the resources can only belong to a single cluster - there is no cluster inside of a cluster thing.
Edit - I looked at the link that you posted and I'm not sure why the author stated "•Cluster name for each node". My only guess is they meant each node needs a name and IP (for the node). Otherwise it's not a correct statement, the author should probably be contacted.
Best Answer
It depends on what your goal is.
Technically, SQL Server will work just fine in a Availability Group or FCI installation with a single domain controller. I have this set up in my lab and it works without issue. In this situation, you will be fine.
Generally, the design diagrams you see are splitting things between sites. So you have a DC and a SQL node in each site. If Site 1 fails, you can fail over to Site 2 and still have authentication still work. If you have only one DC, and the site where the DC is offline, then authentication will fail causing issues for your users.