For example, in 1980 they ruled that it could only be applied only if it involves a precise and aggravating factor[0].
Louisiana had a law that made the rape of a child punishable by death and SCOTUS overturned that law in 2008[1].
SCOTUS has consistent narrowed the crimes punishable by death in the past 50 years using the Eighth Amendment as a justification.
[0] https://en.wikipedia.org/wiki/Capital_punishment_in_the_Unit...
That does not change the fact that SCOTUS has upheld the death penalty generally as being a Constitutional punishment.
Yes, although there are many things that SCOTUS ruled constitutional that are just unpopular in this day and age that nobody treads in that territory.
So we and all the states have the choice to just not practice that, for example. Yes, in this case that is unlikely. Still an option.
As far as popularity goes though, the death penalty has been more popular than not for all but a brief time in the 60's[0], and I think it's dangerous to conflate popularity with Constitutionality, or to suggest even implicitly that they should have any bearing on each other.
Bear with me, I am not from the US.
The main way things get before SCOTUS is through the parties appealing the lower court decisions and then SCOTUS deciding to hear it.