Buried in the wiki page for YBCO is a note that REBCO is a synonym of sorts. The superconducting tapes that are discussed in the talk thus would seem to be YBCO.

Yes. Re stands for "rare earth," and yttrium is probably the most common one used for ReBCO, but lanthanum is also used.

They don't need to be that cold to start superconducting, 4.2K is just the temperature you could expect with liquid helium cooling. It would actually be superconducting with liquid nitrogen but the reason why you would still want to go colder is because superconductors have a maximum magnetic flux that they can sustain while still being superconducting. The current travelling through a superconductor itself also contributes to the magnetic flux so even though it's superconducting there's still a limit as to how much current you can pass through a conductor and how high of a magnetic field you can create. This limit is dependent on temperature though, so you definitely want to use liquid helium so you can create a much stronger magnetic field.

Right. To expand: Maybe someday we can use liquid hydrogen or nitrogen (or even water, depending on progress with hydride superconductors) for these reactors, but a big reason they’re able to make this more compact reactor work is the much higher critical currents/fields that these high temperature superconductors can handle when cooled far below their critical temperature. See: https://fs.magnet.fsu.edu/~lee/plot/plot.htm