Temp table is stored in tempdb until the connection is dropped (or in the case of a global temp tables when the last connection using it is dropped). You can also (and it is a good practice to do so) manually drop the table when you are finished using it with a drop table statement.
No, others cannot see your temp tables if they are local temp tables (They can see and use global temp tables) Multiple people can run commands which use the same temp table name but they will not be overlapping in a local temp table and so you can have a table named #test and so can 10,000 other users, but each one has its own structure and data.
You don't want to generally look up temp tables in tempdb. It is possible to check for existence, but that is the only time I have ever referenced tempdb directly. Simply use your temp table name. Example below of checking for existence
You name temp tables by prefacing the name with # (for local tables the ones you would use 999.9% of the time) and ## for global temp tables, then the rest of the name you want.