Yes, a red back light with 3 brightness levels. The X4 seems like the right keyboard for you, FPS games and the X4 go well, most other keyboards allow 5 or 6 key pushes, the X4 has over 20, so you can wasda to your hearts content.
Technically, it is mouse correction, and whether the correction is high or low. Logitech calls it angle snapping on mice such as the G500, you can turn it off in the software. But at the same time, I know that this is not off, rather the lowest comfortable setting that Logitech feels it should be. Steelseries are actually about to launch a new mouse that eliminates the prediction all together,or so they claim, they also have the Xai, which uses the same sensor as the G500. The difference between something like the Xai and the Microsoft Intelli is so small, that most people cant actually distinguish between them.
Skipping is again, something that people will call mouse companies out on, but the G500 sensor, for example, has a decent top speed, which if your mouse doesn't, it will stop tracking or spaz out. Optical mice generally have good top speeds in this area, though something like the MX518 is limited to 125hz, so you will need to change that to 1000hz in order to get the best tracking and lowest level of prediction, the Zowie EC2 is a good example of this in action, that is the closet you can get to a Intelli without being one.
High DPI to a point is just marketing, but adjustable DPI is used but a lot of top gamers and has real benefits, having a low in game sensitivity and high DPI is common place.