Hi, and welcome to the world of caving! A "register" is a canister of some kind that is posted in the cave to allow people to say "I was here," without defacing the cave with graffiti. It is usually a PVC tube chained to the wall, and if you can get the darn things open, they are usually full of soggy papers with a bunch of signatures and their witty comments. Sometimes the registers are difficult to locate, and sometimes they are removed for whatever reason.
The circles with the numbers indicate ceiling heights, generally in feet.
Here is a website that shows the commonly used cave map symbols:http://www.carto.net/neumann/caving/cave-symbols/uis_signatures_english.pdf