Whether you can install tandems depends on the panel. Some panels are not listed for their use. If you look at the panel label and see a number that ends with 3040 it could use tandems. If it is like 3030 you could not. If you post the panel model and number someone could give you a more definitive answer.
Yes, a tandem is 2 breaker internals inside of one case. Your assumptions are correct.
Yes, a tandem is 2 breaker internals inside of one case. Your assumptions are correct.