I understand that mode S transponders have unique identifiers, thus
eliminating the need for beacon "codes" as we use them today. Is that
right? Do people flying with mode S today still need to use beacon codes
for compatibility reasons?

TIS travels via the mode S data link. TIS-B uses ADS-B's data link. So
no mode S transponder is required.

So, in an ADS-B world, what's the story on transponders. Is mode C still
required? Does the ADS-B UAT replace a transponder? Something else?

Thanks...

Andrew