I can configure an IP address and subnet mask for the adapter (i.e.,
the hardware itself?), such as 192.168.56.1/24. I think this means
the adapter will handle any requests to the IP address range of
192.168.56.0-192.168.56.255.
No, that is incorrect. An IPv4 address has two parts: the network and the host. Each device interface will have a host address on the particular network. An interface with the 192.168.56.1/24 address means that the network (the first 24 bits of the address) is 192.168.56 and the interface address (the last eight bits of the address) in that network is 1. The interface only responds to that particular 32-bit address (excepting broadcasts and any multicasts for any groups to which it is subscribed).
I am not sure what 192.168.56.1 refers to specifically; won't anything
starting with 192.168.56 go to that adapter anyway?
No, only packets destined to the full 32-bit IPv4 address will be used by that interface.
On top of that, if I configure DHCP on my virtual network adapter, I
again set an IP address and subnet mask but also a DHCP range. For
example, an IP address/mask 192.168.56.100/24, then a range of
192.168.56.101-192.168.56.105. The DHCP range makes sense because it is the pool of addresses that the DHCP can assign, but what is the
DHCP IP address for?
I do not really understand what you are doing with this. DHCP (Dynamic Host Configuration Protocol) is used to assign information to an interface, like IP addressing, a gateway address, DNS addresses, etc. If you are running a DHCP server on your host, then you would assign a range of addresses for a scope that other hosts use, and the DHCP server itself would require an IP address because it is a host on the network. If you are using DHCP to configure your interface, then you would not normally configure an IP address manually on the interface.
I am unclear where the gateway IP address (which I thought would be
the address with the bits after the mask set to 0, traditionally) fits
in here.
No, that would be the network address, not a gateway address (except as in the IPv6 router anycast address). A gateway (usually a router) is a host on the network that know how to reach other networks. A gateway address is just another a host address, and it could be any host address on the network that gets assigned to the gateway.
You can learn more about IPv4 addressing in this two-part answer.
