Does the radiation from the sun/sun's radiant energy change over time?

Then, several billion years from now (I think around 6), the sun will burn off all of its hydrogen

5-6, yeah. Note that it's only the hydrogen reserves in the core that will burn off though, not the entirety of the Sun's hydrogen. The H in the outer layers will almost entirely be lost to space in the late stages of its evolution, rather than being fused.

Once it burns through its helium there will be a brief stage where it combines lighter elements into heavy elements as it dies

This specifically occurs through s-process nucleosynthesis. This is a byproduct of the high neutron density in the interior of ageing asymptotic giant branch stars (old "red giants" that have exhausted their core helium reserves), and doesn't occur at nearly the same frequency as sustained core fusion does, so it's not like a sizable chunk of the core will convert to heavy elements like when the core's helium converted to carbon and oxygen. It's more like one or two individual neutron capture events every few decades or so.

eventually completely shedding its atmosphere in a nova, leaving behind a small, hot white dwarf, the end of the white line.

Not quite. A nova is an event where hydrogen falling onto the surface of a white dwarf (e.g. from a companion object, or a surrounding accretion disc) gets hot enough to start fusing into helium, causing an eruption on the surface of the WD. What you're thinking of is a planetary nebula.

There's also some very complex evolution that occurs during the gap between the planetary nebula formation and becoming a bona fide white dwarf that is often overlooked, even by professional astronomers. A percentage of former Sun-like stars can actually experience very late helium flashes, either while approaching or just after falling onto the white dwarf cooling track, which can briefly return them to the asymptotic giant branch. Stars that are in transition from the "born-again" AGB stage to the white dwarf region include hydrogen-deficient objects like PG 1159 stars and [Wolf-Rayet CSPNe (planetary nebula central stars, formerly with Sun-like masses, that show similar spectra to high-mass Wolf-Rayet stars, and which appear to be the immediate precursors of the PG 1159 stars).

That HR diagram also isn't entirely accurate - the Sun is expected to remain at peak luminosity for a while longer after leaving the RV Tauri variable stage, so it should cut across the main sequence right up in the extreme blueward region, not at ~9,000 K like that diagram shows. Objects like PG 1159 stars and newborn white dwarfs are very hot, with the hottest being ~200,000 K or so. This diagram is probably a bit more accurate, although I think its peak luminosity is predicted to be more like 4-5,000 times its present value (which is consistent with some of the brighter RV Tauri variables out there), not ~1,000 like shown there.

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