Why am I seeing red dots on my LCD screen?

Sadly, this sounds like bump underfill cracking on the GPU. It will get worse and worse and eventually it will destroy the machine. There is one way to prove it conclusively -- does the problem tend to get worse when the laptop is cold and get less serious as the laptop warms up? (Or did it start out that way?) If so, it's definitely bump underfill cracking.

As the GPU heats up and cools, the components that join the GPU die to the motherboard expand and contract at different rates, creating stresses on the connections between the GPU and the motherboard. Eventually, the material starts to crack, leading to intermittent connections. At least at first, the problem tends to be worse when the laptop is cool because as it heats up, the expansion tends to force the bumps into better contact with the underfill.

It's very unlikely to be an issue in the display because the red dots move with the windows.

For me it was only a bad connection between the monitor cable and graphics card.

I've just had an identical problem. It has been solved by fixing (adjusting/reconnecting) the monitor cable to the graphics card, as suggested above.

1. The GPU overheats (dust and/or fan problem and/or faulty component) or

2. The balls on the GPU are unstable due to lead free solder (expand when hot and cool down when PC turned off – thus when cold the problem hits you in the face but when heated up sometimes not so bad)

Usually, it's 2). Unless you are competent enough to reball (basically heat up each point/ball to reflow the solder) better replace the graphics card / GPU.

Had a similar issue after replacing a screen. The pixels almost flickered as well. I read this some where, but just pushed on the screen and bezel. Basically where the connector is on the back screen (inside of the screen). Fixed it, I must have had the cable a little loose.