How far can apartments go? I mean is banning tattoos any different than banning african-americans, asians, etc?

The fact that it is a personal choice does make a difference, but not one that should matter.

Having tattoos does not make you a bad risk or a bad tenant.

So what if instead of tattoos they said must not shop at walmart?

What if they said must not have unmarried sexual relations?

Must not have dreadlocks? Must not.. well you get the point

Conversely, do apartment landlords have any say who can or cannot rent a room? Or, by government enforcement, they have to accept anybody who comes around with an application, with a room being available?

If a tenant doesn't like the living conditions, they can leave, or better yet, not move in at all. A person can and should shop around for a place that suits them best.
Shouldn't landlords have at least some say in who can or cannot rent?

Honestly, it's their loss if they reject people on frivolous reasons.