Nowadays we have multilayer switch and pure layer 3 device.
Do "pure" layer 3 device like a router use layer 2 protocol ? Do they require layer 2 header to operate ?
If so, what are the differences between multilayer switch and pure layer 3 device ?
Any service in the network layer needs to use the data link layer. In turn, any service in the data link layer needs to use the physical layer.
If the router is a Layer 3 device, does it even support Layer 2 forwarding (like a switch does)?
No. Each router port belongs to a different IP subnet/L2 segment and you don't want MAC-based forwarding or broadcasting between those.
Do "pure" layer 3 device like a router use layer 2 protocol ? Do they require layer 2 header to operate ?
They use and require layer 2 but don't forward based on L2 addresses.
If so, what are the differences between multilayer switch and pure layer 3 device ?
A multi-layer switch provides both, layer-2 (frame-level switching within an L2 segment/subnet) and layer-3 services (packet routing between subnets). A pure layer-3 device only provides packet routing.