This is a great time to direct folks to the
Cordor Information Thread, which has some pretty basic info on most all of the individual districts and their inter-district politics. It's reasonably up to date. There's also the
Banishment List, which seems a little more outdated but includes some info you may be looking for.
But, to summarize, yeah. The West District tends to be separate from the rest of the city in many political matters. One of those is who it permits in its gates; drow, kobolds, and other typically-banned creatures are permitted there. However, the populace is almost entirely human, and non-humans tend to get rather dirty looks on frequent occasion. It is a more lawless place, so... While a drow or kobold may be able to get in, they are also more likely to get, say. Lynched. By a mob. Or something. (Though this is rare unless they cause trouble. Still, nonviolent prejudice is a thing.)
Partly related to this is the fact that Central Cordor is quite well-guarded right now, so don't think you can sneak your way into the rest of the city without trouble.
The rest of the city bans kobolds with extreme prejudice, and most drow, except for in the North District. Many churches, cults, and other organizations and individuals are banned as well, and all of those bans generally apply to every district except for West.