Is it possible to connect an external GPU via Ethernet?

No.

Chances are very slim that your laptop has a 10 gigabit ethernet adaptor - its uncommon on high end desktops. You'd also need some clever hardware to translate between PCI-e to ethernet, and the latency, oh the latency. You have a device which is typically put as close to the processor as possible with 16 dedicated pci-e lanes right into it, and you'd be adding a ton of latency.

Such a device would essentially be a video card, plugged into a PC, rendering what you want to play on the video card, compressing it, and sending it over as video.

Newer laptops may be able to do this with thunderbolt, but that's a whole different protocol, with the necessary clever hardware available.

Connecting a GPU by Ethernet is like connecting your PS3 controller to the PS/2 keyboard port: sounds like it's trying to fix a problem but the solution turns out to be completely outlandish and impractical. (Granted, if you can pull it off, you'll gain more popularity as that one crazy guy.)

That said, I advise you not to try to do this because the interface simply doesn't exist. You'd have to go through a lot of hoops to get your card to accept low-level commands through high-level interfaces and then bring back a video output. The latency would be absolutely horrible, unless you are willing to pay thousands and thousands of dollars for an enterprise-grade solution that lets you do this for one reason or another.

However, don't fret. You may still be able to connect your GPU to your laptop. For instance, if you have an ExpressCard slot, you can use an ExpressCard to PCIe adapter (along with a power supply unit) to have a seamless experience.

However, there may be compatibility issues due to the operating system's inability to set up resources for the PCI connection to the GPU, so browse around and make sure it's compatible before you buy anything. Consult guides such as PCWorld's eGPU tutorial to see if you have everything you need to use a desktop GPU with your laptop.

Regardless, do not expect 100% performance. Remember, it's a laptop.

Probably not at commodity pricing for another few years. The current offerings are all high-end, using 10/40 Gbps connections and Nvidia Tesla cards. Even then, it's not directly usable for gaming or graphics, but more for CUDA processing.

e.g. http://www.chelsio.com/gpudirect-rdma/