Are specific impedances for high-speed signals maintained beyond the PCB connectors?
But what happens to these impedances beyond the connectors which the controlled-impedance traces are routed to? Are the impedances maintained?

Many times, if not most, yes!

If not, then is it really necessary to design/route the traces with the specific impedances?

Hm, it's usually useful to at least make your PCB traces not work as antennas for the signals on them, so that you at least not interfere with yourself.
But: controlling the impedance of a cable is much harder (it's flexible! The acceptable cost per millimeter might be much lower!) than that of a trace, so that drivers and receivers for external cables do have to be designed with reflections and coupling in mind. But that also gets a bit easier when, at least between the chip and the connector, things are matched.
In reality, most of the things you find that are high bandwidth have a defined RF impedance: HDMI has 100 Ω (with a relatively big tolerance, low cost!), SATA cabling 100 Ω (with lower tolerances), USB3 superspeed pairs 75 to 105 Ω (low cost!), CAT5,6,7 ca 100Ω, Good Ole VGA 75Ω.