Yes, the FAA moves the regs around every few years - keeps instructors learning.
If you hold a FAA ATP SEL then you do not have to take another written. The Airbus checkride will also serve as a ATP MEL checkride.
There is an "or" at the end of 61.64(2)(iv) so you can read "or" at the end of (i),(ii),(iii) and (iv). So if you meet the requirements of one of those paragraphs, you can train in a sim.
There are places in the regs that say "issued under this part" when the FAA is requiring something done under US rules. Since 61.64(2)(i) just says "type rating" and not "type rating issued under this part" I would think a JAA type rating is OK. The idea is that while people say the sim flys like the airplane, there are things that are not realistic in the sim. The FAA wants you to have some real world experience in turbojets before you complete a type rating in a sim.
Since you are not working for an airline, you would complete your Airbus training at a Part 142 school. Part 142 covers type ratings in sims. The 142 schools in the US have many foreign students and can answer your questions about FAA and TSA requirements.