What kind of host OS needs to be installed to run the VHD made from the previous server's hard drive?
Since you want to run that instance as a virtual machine, you can install any hypervisor OS you prefer, be it Windows Server 2022 with Hyper-V role, Hyper-V Server 2019, VMware ESXi, any Linux KVM flavor, Proxmox VE or XCP-NG. Not all of those hypervisors support VHD(X) files as virtual disks, so you may have to use a free V2V tool https://www.starwindsoftware.com/starwind-v2v-converter to convert your VHD(X) file to a supported hypervisor.
Can it be activated with the same license key as the guest OS (the one activated in the VHD that was created from the previous server's hard drive)?
That primarily depends on the license you currently have. In most cases, you license the host, not the OS. That means you must check whether your current license is not bound to the old Dell PowerEdge R250 server and/or if you have transfer rights. Otherwise, you will need to purchase a Windows Server license for the new server as well.
If not, and we only get to migrate the activation key from the previous server's hard drive to be used in the guest OS on the new hard drive, does that mean we have to pay for another activation key for the host OS on the new hard drive?
That depends on the host OS you intend to be using. Most of the hypervisors I mentioned above are free and do not require any license.