VDS usually corresponds to a dedicated hardware server, meanwhile VPS is typically a VM.
VPS can run WireGuard as long as its kernel has enabled the relevant modules.
It's usually easy to achieve on a KVM-based VPS where you can install virtually anything.
The once I've had in the past did, yes.
But then again I've only used the $10/y ones ,)
Most VPSes you can select what kernel you want to boot... Like in Linode or Digital Ocean both allow you to select kernels, including your own kernel you install from a distribution.