The FAA only has one rating for 737s, covers them all from -100 through -900. Most of the 737 schools use a 200 sim for their basic course and then you can do differences for the classics. Or pay more and do it all in a classic sim. Yes, you'll get a FAA certificate. There is no "conversion" above the private level in FAA land unless you hold a Canadian license. You'll have to pass the written, hold a FAA medical and get through the TSA mess.
As to where to go, google brings up half a dozen schools when you enter 737 type rating. Hard to say what's best for you.
Last edited by MarkerInbound; 2nd Sep 2012 at 05:16.