In practice, many of the cities with the highest housing costs are now strongly anti-infrastructure and anti-transport anytime it involves cars. (What good is a self-driving car if it's getting stuck in the same-or-larger traffic jam everyday?). Cities like this problem because it adds to the inflation of their own property values, and they get to pretend they are being "greener" or "safer" (even though self-driving EV cars eliminate the majority of those issues).
So I suspect that in practice, self-driving cars won't lower housing costs much in most major markets. It could help smaller cities, but they already struggle less with housing costs in the first place.
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The biggest changes will probably be felt elsewhere. Such as by professional drivers (school/city bus drivers, truck drivers, taxi/uber, etc). I also suspect some spill over against flights. It might soon be cheaper for a family of four to rent a self-driving Chevy Bolt to drive from ORD to MSP or ORD to ATL, than to fly there. Especially since, once you land in MSP/ATL, you'll need a car to get around anyway.
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Wealthy people will continue to be mostly unaffected, as they can already afford to live where they work, and can already choose to have no meaningful commute, anytime they desire to.
These cities would still be better off with SAE Level 4/5 and inter-communicating cars, which should reduce the frequency of traffic jams significantly. Traffic jams are almost ALWAYS caused by human error at some point in the chain.
Sometimes, sure. But there are a lot of key areas where urban roads / freeways are simply past usable capacity (even with smarter self-driving cars communicating together).
Most major cities have experienced significant population growth in the last 10 years, and have also built little-to-no new freeways or additional car capacity in the last 10 years. If I had to guess at the future, I would predict both of those trends to continue.