Couple of different questions here:
Gaming Laptop: Mobile processors, mobile GPUs. Some mobile GPUs are slightly disabled full desktop GPUs running at a lower clock frequency. Some of the available processors are nearly as fast as their desktop counterparts, and starting this generation. Intel is offering a few actual desktop CPUs for mobile.
NUC: A Nuc is essentially a laptop without a battery, screen, keyboard/mouse. These are extremely portable if you have carry wireless peripherals. You can even get a USB display or at least a very small display to carry around.
SFF - Small form factor computers are available. These are usually a pairing of a laptop CPU with a discrete desktop GPU. Not sure I know of any recent ones, not much of a market for them. They had a brief renaissance with the introduction of Steam OS.
ITX - A small computer you can build yourself. 8" square motherboard with a 10.5" or smaller full GPU. Almost any class of hardware is available up to workstation class processors and dual GPU cards. (Though you are limited by power supply size, some have SFF supplies, others support full ATX supplies, but they are bigger of course)
Plenty of portable cases out there. Look up LAN party cases (they generally have carrying handles)
Once you have narrowed down what you are after, and decide on a budget, maybe we can put together a build.