Even pretty small ones do. My wife has worked at Amazon as an account manager both for 1P and 3P sellers, some of those don't even make $100K a year on Amazon but still have an internal contact.
Depends on the category. My brother runs an Amazon store that nets more than that but his category is one of the strictest on the site and he gets no support.
I mean those programs aren't free for 3rd party sellers, so if he doesn't pay I'm not surprised, but it likely doesn't have much to do with the category.
What I've heard about having the "Premium" (pay) version of Amazon account manager: It's just another layer of the same of the usual awful seller support. Since the "manager" can't actually do anything, having one is worse than not having one.