Is it possible at least in principle for something to have zero resistance to electric current but not otherwise have the properties of superconductivity? Or are those other properties such as the Meissner effect more or less inevitable consequences of zero resistance?
Yes. A perfect diamagnet would have zero resistance, without demonstrating the Meissner effect. Within a perfect diamagnet, the magnetic field cannot change. In a superconductor, the magnetic field is expelled on the instant that superconductivity is attained.
http://hyperphysics.phy-astr.gsu.edu/hbase/solids/meis.html#c2
A dense plasma is an almost perfect diamagnet, and a gas that is heated to plasma with a magnetic field within it will trap that field as shown in the cite above.