Microsoft Remote Desktop via Ethernet Cable

While the latest version of Microsoft RDP protocol supports RemoteFX which is an implementation of a virtual GPU, "game play" is a very broad term.

RemoteFX vGPU: the ability to present a virtualized instance of a physical GPU into multiple Windows 7 virtual machines. This provides VMs with access to the physical GPU, enabling hardware-acceleration for rich graphics scenarios such as 3D rendering and game play.

Your game (never mentioned) may not be 3D intensive (maybe it's full contact tic-tac-toe). Then again, it may be the latest release of Halo. Will the game "play"? Yes. You should be able to play it remotely.

Will it lag?

It depends. My rule of thumb is that video and games are best played locally and not over a shared screen.

I have a Linux box connected to my Mac via gigabit ethernet cables, a switching box and VNC. The VNC connection is still so slow it struggles to run media-rich websites. I suspect the easiest option for you would be to use a monitor switching box that would allow you to swap control of the main monitor, keyboard and mouse between the PC and the Mac.