JP, Not just the Klein-Gordon equation, all the fundamental equations of physics are time-symmetric. Given any solution, you can produce another equally valid one by replacing t → -t. However that does not mean that every solution is physically acceptable. In addition to satisfying the equations, there are boundary conditions which must also be satisfied. Time flows in one direction and one direction only -- this is an observed fact, and to satisfy it we impose "retarded" boundary conditions on our solutions.
This is not correct. Antiparticles propagate forward in time just like normal particles. (What about particles that are their own antiparticle, like photons. Which way would they propagate?) The Klein-Gordon equation has both positive and negative frequency solutions. An early interpretation of it assumed that negative frequency meant negative energy, and a negative energy solution would indeed represent a particle traveling backwards in time. But within a few years it was recognized that the interpretation was wrong, and such solutions were replaced by antiparticle solutions traveling forward.
The Poincare Recurrence Theorem is a statement of ergodicity, and applies to closed systems with constant energy. It does not, for example, apply to the expanding universe.