MT,
All of the above is correct. There are two additional steps you will need to complete before you start training. First, the FAA will have to verify your licenses with your CAA. The FAA says it can take 45-90 days to complete this verification but I've heard it normally can be done in a month. Secondly our friends at the TSA will have to do a background check on you. Since you are getting an A320 type rating you will be training at a FAR Part 142 school. All the 142 schools I know see this all the time and will be able to walk you through the steps required.