Putin's goals are not peace, they are restoring the soviet union (his vision of what it should have been, not what it ever was). All call that crazy. That should not be confused with insane: he has a sharp mind, just he has turned it to things that I consider crazy to consider.
Ukraine matters to NATO because they are turning more and more to our classical liberal values of freedom, and that is something NATO wants to encourage in general. (in general - note that Turkey and Hungary are part of NATO that don't want to encourage this, and there are other countries not sure)
We can only guess why Russia/Putin thinks Ukraine is existential to Russia. Our best guess is because they were a historical part of the Soviet Union that they are trying to restore. Putin cannot fulfill his vision without Ukraine. NATO countries like Latvia were also part of the Soviet union and so we expect they are next.
Ukraine matters to NATO because they are turning more and more to our classical liberal values of freedom, and that is something NATO wants to encourage in general. (in general - note that Turkey and Hungary are part of NATO that don't want to encourage this, and there are other countries not sure)
We can only guess why Russia/Putin thinks Ukraine is existential to Russia. Our best guess is because they were a historical part of the Soviet Union that they are trying to restore. Putin cannot fulfill his vision without Ukraine. NATO countries like Latvia were also part of the Soviet union and so we expect they are next.