Washing the garage door can be quite simple. Just use a household detergent mixed with warm water. Lather and scrub the garage door, rinse, and let it dry. Make sure that the type of detergent or soap you are going to use is not too strong to damage the finish of the garage door, especially the wooden ones.
Rusty garage door parts do make terrible noises and this is due to friction. When their surface is completely damaged by erosion, the moving parts cannot slide smoothly. They are actually dragged on other components and produce loud noises. That's why lubrication maintenance is important.
The cost of modern garage opener remotes would depend on the brand and their capacities. The latest models are more expensive than older ones as it happens with all products but they're not really expensive and offer plenty. It would also depend on whether they're multicode remotes, too.