First off, if you have two contrasting opinions from orthodontists you may want to find a third orthodontist and see what the consensus on the extractions seems to be. Extractions should be avoided if possible because they are painful and tend to impact facial structure.
Regarding self-litigating (i.e. Damon System) versus traditional brackets, the type of bracket generally has no effect on the results of the treatment. This makes a lot of sense, since the brackets themselves do not move the teeth, but rather are only there as a means of attaching the wire to the teeth. The wire is the structure which actually moves the teeth into place.
The Damon System has a brilliant marketing campaign, but you should speak with licensed orthodontists to find out how much of it is true and how much of it is advertising technique. Be aware that promotional material for the Damon System does not actually cite any studies or medical publications.
Here's a source I found that explains this more in-depth: http://tdrortho.com/deciphering-the-tru ... ng-braces/