It was cancelled in 1971, so well after the first ICBM was launched but before SALT. It seems like the sort of thing that would be useful in the event of a ground war, so I'm guessing total annihilation became the expected outcome of a war with the USSR?
The W54 warhead for the Davy Crockett was produced between 1961 and 1965, and retirement started in 1967, was finalized in 1971. The idea behind the thing was to repel Soviet tank and troop columns — it is a "force equalizer" in that it lets a very small ground force attack a much longer one.
As to your secondary question, tactical nuclear weapons, which were imagined to be used not against cities but against other local military forces, were deployed throughout the Cold War. (The Davy Crockett was neither the first nor last of this category, in other words.) The idea was that there might be "steps" of escalation and that "all or nothing" would be called as a bluff. Low-yield nuclear weapons were meant to signal, "we're not afraid to start a little nuclear war, so don't make a little war with us."