1. ERAS is the application service. NRMP runs the actual match. They're related but different. Brodmann was being a bit pedantic I think...or confused. In either case, you need to register with both and your rank list will go through NRMP. ERAS doesn't really have "deadlines" per se, application deadlines are set by individual programs with Surgical programs generally earlier than IM/Peds/FM.
2. Application timing does matter but it's not really that straightforward. If you have a solid, but incomplete app, applying early doesn't really hurt and might help (get an IV offer you might not otherwise get and really impress them on interview day). If your app isn't stellar, waiting a little bit (for an awesome LOR, for a transcript that includes all honors, for a 250+ Step 2, etc) is probably beneficial. The caveat to that is that since this is a rolling process and programs only interview a certain # of applicants, the longer you wait, the lower the chance of getting an interview.
My personal thought is that, as long as your app is relatively red flag free (arguments exist as to whether IMG/FMG status is such a flag), you should apply early. The filters that PDs use don't throw out your app, it just gets hidden. A week later, when they run it again and your sub-I grade or Step 2 score is in and you now meet their interview/review criteria, your app rises to the top of the pile and you're golden. If you wait until the day before the program's deadline OTOH, you risk all of the IV spots being given out already and you wind up on a wait list.