Axle slingers are pressed onto the inner axles. The ride right next to the end of the axle housings. You are correct in your thinking - they keep "big stuff" out of your axle tubes. They do not seal, however. I know many people who do not run them because they feel that while they do prevent big stuff from getting in, they DO let small stuff in, which can easily become trapped in your axle tubes.
The differential is sealed with the inner axle seals, which reside right next to the diff.
IMO, leave the axle slingers off. Most things that find their way in there will be able to find their way back out. When you are changing wheel bearings or repacking bearings, just be sure to give your tubes a good cleaning.