Why are optical mouse LEDs red?

It is not necessary that it be red. Any color can be used, even invisible (infared). The aesthetic design usually does not have to change, but internally it may due to the different type of laser diode. Red lasers are cheaper and readily available. Other colors do exist in consumer products currently, the most popular alternative is green.

Is it necessary that a mouse's LED is red?

No.

What other colors can/can't be used?

Any colour can be used.

Does using another color changes the main design of the mouse?

No.

Does it makes manufacturing mouses harder or more expensive?

Both.

For more information, see this Wikipedia article.

Red colour has the most dispersion compared to other colours in the visible spectrum, violet having the lowest.

Dispersion means how much light can 'spread', 'disperse', 'scatter', etc. It is the same reason that traffic lights for stop are red. This is so that the sign is visible even from long distance & with different viewing angles.

For similar reasons the mouse may be using the red colour. Not everybody uses the mouse on perfectly flat surfaces!