Irrc, the LED is driven by an output from the image sensor, dim until it detects movement and then bright for precise imaging. On some models the LED is driven directly by the output pin and on others, via a driver transistor. A few more things to check before condemning the LED completely. Wasedadoc's suggestions should eliminate the LED anyway (unless it has failed short).
Regarding question 2, the mice that do not emit light (and have no ball), I think these probably use an IR LED.
I think desoldering the LED and just testing it is the fastest way to tell so I did just that.
Actually, since I had the spare mouse that worked, even while LED is still on the board I was able to test it and it does light up(when testing with multimeter).
On to Q2, I learned that most optical mice emit red light underneath, they use transparent 5mm led that emits red light because I guess red light works the best.
Like you said, some mice do dim until movement is detected, most likely this extends the LEDs life too, so good feature.
I also have one mouse, which I will try to fix next (Razer deathadder) which is optical mouse and for some reason uses purple light LED, which can not be seen by naked eye, you have to look at it trough your smartphones camera.
And yes, finally there is IR mouse.
I have one too, underneath there is a laser warning sticker.