Is it possible to sand/polish LEGO bricks? I have an old white and rare brick which got scratches and dirtying which I cannot wash away. Any ideas on how to clean it? I thought about fine sanding or polishing but I am afraid that I damage the brick.
It is definitely possible, since ABS is a plastic that can be either mechanically or chemically affected. Cleaning bricks is a well-studied field, you will find useful information regarding this topic all over the internet, including here.

Polishing LEGO bricks is less heard of, but can be done either mechanically (another link) or chemically (another link). Note though, that any kind of polishing includes either removal or rearrangement of material, so the chance of damage is definitely not zero in any method. I'd suggest starting with the gentle and precise methods (file, hobby knife, fine sandpaper, etc), and only progressing to more drastical methods (acetone) if they fail.

When using acetone, be especially careful, as the vapours can't be easily controlled, and they are flammable (can burn down your house), poisonous (can kill you and your family), and can even melt the entire brick if given the chance! Of course the dangers depend on the amount and ventilation, but some of the above links depict clearly dangerous settings, uses and amounts.