Liquid crystal displays work by using liquid crystals where the direction of alignment of the molecules in a plane rotates as you go from layer to layer. Usually this is done with twisted nematic crystals, however cholesteric LCs can also be used to achieve similar effects (I don't think smectic LCs are useful here as they don't align).
To create a liquid crystal cell, twisted nematic liquid crystal is filled between two crossed polarizers1, and a backlight is sent through. The important thing about liquid crystals is that the specific rotation is easy to control by varying the potential difference (more potential difference leads to more anisotropic disorder).
Now, if you make a grid of these and put color filters on them, we get an RGB LCD display.
What happens when you press on them? You change the concentration/length of the liquid crystal, as well as making it bulge out slightly2.. This changes the optical rotation it induces, causing the amount of light that comes out to change (as the amount of light is dependant on how aligned the rotated light is with the second polarizer)
I'm not too sure of the exact reasons behind the phenomena you list, but I can guess:
Firstly, the color changes are caused because each type of crystal cell is affected differently by the pressure. I think there's a difference in concentration between the different color cells to better suit human eyes. When I do it to LCD displays accessible to me, I only get a darkening, not a color change, so it seems to not be a universal feature of LCD displays. If the rotation of each color cell is affected differently by pressure, then pressure can easily cause color distortions as each color will not darken by the same amount.
I think the ripples are caused by high pressure where you change the optical rotation of the inner cells (the ones closer to your finger) so much that they go through a at least one full 360o rotation, leading to regions of both bright and dark light.
The persistence is because liquid crystals are viscous — they have to be, as they consist of long molecules. They seem to align quickly to potential differences (potential differences align the molecules without moving the liquid), but when we add pressure (where the liquid itself distorts) I think it takes some time to bounce back. The ripples move due to this same persistence.
1. If you're ever at a 3D movie, have a look at your phone screen with the 3D polarised glasses on. It looks pretty cool!
2. If this did not occur, then the rotation would stay the same as $\theta\propto lc$ and $lc$ does not change if there is no bulge.