Levashov,
Actually each training center abroad depends on a center in the USA. For example, I did a training on Falcon 900 in Dubai, their training facility is under Dallas. Contact the training center and ask them. They will give you the USA facility to chose from on flightcandidates.org
Indeed you will not need finger printing again. TSA approval will take you 3 or 4 days.
Once you fill out your TSA form on the website, you will need to give your username and password to the training center so that they can process your application. Therefore no matter what, you need to get in touch with your training provider ASAP.
Cheers