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.