For 400$, you won't get much. You'll get a pretty weak laptop. Decent laptops start at 800$. Look for Intel i5 or i7 processors with decent clock speeds. Make sure the model name of the laptop's processor does not end in U. U indicates that the processor is ultra low voltage and that will slow down your performance. If the processor in question does not end in U, you should be fine.
If you can get an i5 or i7 processor with HQ at the end of the model name, awesome. You're set with the processor part. For graphics cards, Intel HD can be decent but it's not optimal performance. Look for laptops with Nvidia graphics cards. Laptop versions of Nvidia processors will end in M, meaning mobile.
For example, the lowest Nvidia graphics processor in a laptop is the Nvidia GeForce GTX 760M, I believe. Most of the time however, high performance laptops will have better Nvidia graphics cards.
A good one is the Nvidia GeForce GTX 870M. It should be able to smoothly render games at decent resolutions and at high resolution with some compromised settings (either of the game or of the card).
You probably won't need to check the amount of RAM because most laptops today already have a minimum of 8GB. Do check the type of drive it has, however. Most hard drives today are fine, but if the laptop happens to have an SSD, nice. SSDs are fairly speedier than hard drives. Some laptops are hybrid, meaning they have both. Don't worry about drive size. Most default sizes are already good enough.
There are other factors to consider, such as touchscreen and speaker quality, but I'll assume those are things you're willing to sacrifice, especially if you already have a headset.
On a final note, always check how hot the computer gets, especially if you can use the functional display at places like Fry's Electronics or Best Buy. If it gets kind of hot whilst idle, you might want to either ditch that laptop or get a good cooling pad for the laptop. However, cooling pads that can actually cool down a laptop can cost up to 80$ I believe. This is my advice. Hope it helps.
Edit: I can't recommend any specific gaming laptop but I hope this guide helps. A gaming laptop should also be good for demanding tasks such as video editing in case you were wondering. Good luck!
Short amount of time is not really useful for low performance computers. Programs will fill up cache folders really quickly again, rendering your suggestion another nuisance rather than a solution.
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